- Email: [email protected]
Effects of entrainment and deposition mechanisms on annular dispersed two-phase flow in a converging nozzle
PERGAMON
International Journal of Multiphase Flow 25 (1999) 321±347
Eects of entrainment and deposition mechanisms on annular dispersed two-phase ¯ow in a converging nozzle S. Ho Kee King *, G. Piar Laboratoire d'EnergeÂtique et de Thermique Industrielle de l'Est Francilien (LETIEF), Universite Paris XII-Val de Marne, IUT de CreÂteil, Avenue du GeÂneÂral de Gaulle, 94010, CreÂteil CeÂdex, France
Abstract This paper is intended to establish a simple three-®eld model, which gives good agreement with a converging nozzle data previously obtained. The problems encountered in developing a mathematical model for two-phase ¯ow are numerous and generally complex. More particularly, the main diculties concern the closure relations generally correlated empirically and which govern the various transfers at the interfaces and walls. Our methodology consists in taking into account at ®rst the limits of the annular dispersed con®guration before reaching step by step its complete description. Such a procedure underlines the necessity to describe the entrainment and deposition mechanisms in order to get a better annular dispersed two-phase ¯ow understanding. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Two-phase ¯ow; Air-water; Annular dispersed ¯ow; Converging nozzle; One-dimensional model; Closure laws; Entrainment rate; Deposition rate; Entrained liquid fraction; Onset of entrainment
1. Introduction The study of two-phase ¯ows is necessary to evaluate or optimize the performances of many industrials systems currently encountered in chemical, thermal or nuclear engineering. For example, in petroleum industry, where ¯uids are generally in pressurized tanks storage, the security of industrial equipment relies on the correct sizing of the safety valve to prevent for an eventual accidental rupture of a lique®ed tank, or the protection of chemical reactors against thermal runaway. * Corresponding author. Tel.: (33) (1) 45 17 18 41; Fax: (33) (1) 45 17 18 42; E-Mail: [email protected] 0301-9322/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 9 3 2 2 ( 9 8 ) 0 0 0 4 3 - 3
322
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
The purpose of this paper is to investigate how such a ¯ow develops through a converging nozzle. Owing to the present state of knowledge, we are only interested in developing onedimensional (1D) ¯ow models. Droplet moisture exchange is an important factor for mass transfer of annular dispersed two-phase ¯ows. Such a ¯ow is characterized by a central gas core containing droplets ¯owing with a velocity more rapid than the liquid ®lm on the pipe wall. Its complete modeling is based on writing the conservation equations of mass, momentum and energy for each one of the three-®elds : liquid ®lm, gas and dispersed droplets. The problems generally encountered in modeling two-phase ¯ows are in establishing analytically the transfers between the ¯uids under consideration and the walls, on the one hand, and between the ¯uids themselves at the interfaces on the other hand. The laws giving these various transfers allow one to close the conservation equations system and are so called closure laws. Among these laws, the more dicult to specify are the deposition and entrainment rates not yet as well-known as, for example, ®lm-gas, gas-droplets interfacial friction, or wall friction. Our work is essentially based on air±water experiments performed at CEA/Grenoble by Lemonnier and Camelo (1993) and Selmer Olsen (1991) on the FOSSEGRIMEN ¯ow loop, with an annular liquid injection mode. This loop is designed by Selmer Olsen in order to produce annular dispersed critical air±water two-phase ¯ow. The experimental data are collected from a vertical downward converging nozzle with a 10 mm diameter long throat : data of pressure and ®lm thickness pro®les, maximum gas mass ¯owrate, size and velocity of droplets at the nozzle exit. The experimental setup and the techniques of measure are described in detail by Selmer Olsen (1991) and Lemonnier and Camelo (1993). In a ®rst step, two simpli®ed ¯ow models are utilized to analyze the experimental data : totally dispersed ¯ow and purely annular ¯ow models. It appears clearly that neither the liquid ®lm or dispersed droplets can be neglected in Lemonnier and Selmer Olsen's experiments. However, it must be emphasized that when the gas mass quality is very high the ¯ow is basically dispersed; so the two-®eld model predicts quite well the maximum gas mass ¯owrate and exit liquid droplet velocities (Lemonnier and Camelo 1993; Ho-Kee-King 1996). Therefore, the three-®eld model must verify numerical results identical to those of the two-®eld model when the two-phase ¯ow tends to become essentially dispersed. At the beginning, the experiments of Lemonnier and Selmer Olsen have been carried out to characterize the eects of non-equilibrium. These authors have already compared their data with predictions of a dispersed ¯ow model (DFM), which neglects the eect of liquid ®lm presence at walls, and where only the dierence of velocities between the phases is taken into account. With the two-®eld model presented here, which assumes the two-phase ¯ow being totally dispersed, any kind of non-equilibrium (mechanical or thermal) is taken into account. In the considered experiments, predictions of such a model have con®rmed that the main source of non-equilibrium is eectively the mechanical's one and that thermal non-equilibrium between the phases can be neglected. The next step consists in a more complete modeling with the ``frozen'' three-®eld model, the whole eects of droplet entrainment and deposition being neglected. So, for the model closure, the main assumption, which is highly speculative, is that a quasi-equilibrium is established between the annular ®lm, at walls and the dispersed gas core from the nozzle inlet. The
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
323
purpose of such a step in our work is to highlight the eects on pressure pro®le of the entrained liquid fraction at the nozzle inlet, more particularly at the throat inlet. In the presented experiments, comparisons of data with predictions of pressure and ®lm thickness pro®les, showed that droplet moisture exchange between the liquid ®lm and the dispersed central gas core would take place in a large majority in the converging part of the nozzle. After this step, we have intended to give a more detailed description of annular dispersed air±water ¯ows in nozzles with the three-®eld model, incorporating entrainment and deposition mechanisms. For lack of suitable correlations of entrainment and deposition rates in literature, our objective is to search for a deposition±entrainment model which provides a good agreement between the three-®eld model's predictions, with available experimental data obtained in a converging nozzle.
2. Limits con®gurations of annular dispersed two-phase ¯ow modeling The two ¯ow models presented in this section are the totally dispersed ¯ow model (two-®eld model ), where the presence of liquid ®lm is neglected, on the one hand, and the purely annular ¯ow model (PAFM ), assuming negligible the eects of dispersed droplets in the gas core, on the other hand. The thermodynamical properties of the ¯uids are de®ned by assuming that air follows perfect gas laws and water to be an incompressible ¯uid. 2.1. Totally dispersed ¯ow model (two-®eld model) Ho-Kee-King (1996) has presented a 7-equation dispersed ¯ow model, namely the two-®eld model, based on six independent balance equations (one mass balance, one momentum balance and one energy balance for each of the two phases), and one equation which imposes the pipe's pro®le (see the Appendix). This last equation appears explicitly in the system because of the following selected independent variables: _ L ; uG ; uL ; HG ; HL ; P; _ G; M M
1
_ L the dispersed liquid mass ¯owrate (kg/s), uG the _ G is the gas mass ¯owrate (kg/s), M where M averaged gas velocity (m/s), uL the averaged dispersed droplets velocity (m/s), HG the averaged mass enthalpy (J/kg), HL the averaged droplets mass enthalpy (J/kg), and P the mixture pressure (N/m2). This model diers from the 5-equation dispersed ¯ow model (DFM) proposed by Lemonnier and Selmer Olsen (1992), only by the fact that it includes the dierences of temperatures between the phases in addition to the mechanical non-equilibrium. The four closure relations to be provided to the two-®eld model are the wall friction from the gas phase t Gp (N/m2), the interfacial friction between gas and droplets t GL (N/m2), the droplets diameter d (m) and the heat ¯ux at gas-droplet interfaces Q_ LG (J/s.m2). For the ®rst three relations just mentioned, we have utilized those of DFM.
324
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
The wall friction is given by the classical Blasius's relation for a single ¯uid ¯ow in pipe: tGp
1 r uG D fru2 ; f 0:0791Reÿ0:25 ; ReG G ; G 2 mG
2
where r and u are the density (kg/m3) and the velocity (m/s) of two-phase mixture, respectively, D the pipe diameter (m) and mG the gas dynamic viscosity (kg/m.s). The interfacial friction at gas-droplets interfaces proposed by Selmer Olsen is correlated as: 1 r d juG ÿ uL j ; Red G : tGL CD rG uG ÿ uL juG ÿ uL j; CD 6:3Reÿ0:385 d 2 mG
3
The droplet size is calculated from the maximum stable diameter according to Berne (1983): 3=5 3=5 3 ÿ2=5 u 25 1 24 s d 4 2 kR 42 p 2 3rL =rG rL which depends of the friction velocity u de®ned as: p u tGp =r;
5
where k is the Von KaÂrmaÂn constant (10.41), R the pipe radius (m), rL the liquid density, and s the liquid surface tension (N/m). For the last closure relation, the heat ¯ux from the liquid dispersed phase to the gas phase is correlated by (Richter, 1983; Dobran, 1987): Q_ LG h~L TL ÿ TG ;
6
where TL and TG are the averaged temperature of the droplets and the gas, respectively (K ), and the heat ¯ux coecient h~L is given by analogy with a ¯ow surrounding a single sphere (Bird et al., 1960): NuL
h~L d 1=3 2 0:6Re1=2 d PrG ; lG
7
where lG is the gas heat conductivity (W/K.m). Figure 1 presents comparisons of predictions with data of maximum gas mass ¯owrate, droplet velocity, and size at the nozzle exit, for all the experiments performed by Lemonnier and Camelo (1993) when the upstream pressure is set to 2 bar and varying liquid mass ¯owrate. The gas mass quality is ranged between 96 and 25%. We ®rst compared predictions of the two-®eld model proposed here (Fig. 1, starry symbol) with still those of the two-®eld model, but setting a heat ¯ux coecient suciently high so as to reach for an eective heat transfer between the two phases (Fig. 1, squared symbol). These last predictions are obviously identical to DFM's ones. Such a comparison showed equivalent results, so Selmer Olsen's assumption in the DFM is con®rmed : the eects of the thermal non-equilibrium can be neglected for these experiments. It should be noticed that for a liquid mass ¯owrate below 50 kg/h the calculated maximum gas mass ¯owrate is slightly higher when the thermal equilibrium is not imposed.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
325
Fig. 1. Comparison of data (circular symbol) with (a) predictions of two-®eld model (starry symbol) and (b) predictions of two-®eld model but imposing thermal equilibrium between the phases (squared symbol). The upstream pressure is constant and equal to 2 bar.
Next, a comparison of predictions (Fig. 1, starry or squared symbol) with data (Fig. 1, circular symbol) showed, according to the conclusions of Lemonnier and Camelo (1993), that: . with a liquid mass ¯owrate value less than 50 kg/h, the maximum gas mass ¯owrate and exit droplet velocity are well predicted; . for higher liquid mass ¯owrates, the discrepancies increase with the increase of the liquid mass ¯owrate; . the droplet size predicted at the nozzle exit is always smaller than the measured drop size, the dierences become more signi®cant with the increase of the liquid mass ¯owrate. This could be the consequence of a phenomena of drops coalescence occurring from the nozzle outlet and the location where the measurements had been performed (at 15 mm apart from the throat end, Lemonnier and Camelo 1995). If now we look at the case where the gas quality is equal to 25%, or the liquid mass ¯owrate is equal to 187.6 kg/h (see Fig. 1), we have also compared the measured pressure pro®le (Fig. 2, symbol) with the pressure pro®les predicted by the two-®eld model. The ®rst pressure pro®le (Fig. 2, thin full line) is obtained with the predicted critical gas mass ¯owrate, which is calculated iteratively so that the downstream pressure becomes equal to 1 bar: its value of 55.2 kg/h is lower than the experimental value (equal to 65.8 kg/h). Thus, it is remarked that, although the downstream pressure is the atmospheric pressure, the calculated pressure drop inside the throat disagrees with the data. The second pressure pro®le (Fig. 2, intermittent line) is calculated with the measured gas mass ¯owrate (equal to 65.8 kg/h). It can be noticed that, up to a certain section, a good agreement is reached with the data and, downstream, the pressure drops so fast that the predictions present a vertical tangent. If we refer to the detailed analysis of Bilicki et al. (1987) on the topological description of the ensemble of solutions, the point oering a vertical tangent
326
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 2. Comparison with measured pressure (symbol) and predictions of two-®eld model for various gas mass ¯owrate values.
should be a turning point ; so the nature of the two-phase ¯ow is critical and the ¯ow is not feasible. Figure 2 also presents another pressure pro®le obtained with an intermediate value of gas mass ¯owrate, and it illustrates the displacement of the critical section towards the throat exit. The two-®eld model can reproduce successfully the experimental results only when the ¯ow is basically dispersed (with high gas mass quality or low liquid mass ¯owrate). In the experiments at an upstream pressure of 2 bar presented in this section, the discrepancies appear signi®cantly when the liquid ¯owrate is higher than 50 kg/h: the maximum gas mass ¯owrate and the exit droplet diameter and velocity are all underestimated, and the pressure drop inside the nozzle throat is not well reproduced. Moreover, since the presence of ®lm liquid is neglected, the two-®eld model is not able to describe the eect of the annular gas±liquid con®guration at the nozzle inlet. The reasons evoked by Selmer Olsen (1991) and Lemonnier and Camelo (1993) in their analysis, are focused on the fact that the liquid ®lm should play an important role on the acceleration of the gas±liquid ¯ow. Then, this could explain the underestimations of the maximum gas ¯owrate and, in the same way, the in¯uence of the gas mass ¯owrate on the pressure pro®le illustrated in Fig. 2. In order to highlight the eects of the ®lm on the mechanical non-equilibrium between the phases, we have analyzed the data with the other limit ¯ow model's predictions, which assumes all the liquid phase as ®lm on the nozzle walls.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
327
2.2. Purely annular two-phase ¯ow model (PAFM) The liquid injection mode at the nozzle inlet is annular, and it is assumed here that the ¯ow pattern remains the same along the nozzle. The assumptions of the previous ¯ow model are preserved, except that the dispersed liquid phase is now neglected and the ®lm thickness is assumed uniformly distributed all around the pipe wall. The system of the equations of PAFM are presented in the Appendix. Concerning the closure laws, the parietal friction t Fp (N/m2) is exerted from the liquid ®lm and the interfacial friction t GF (N/m2) at the interface between the surface of the liquid ®lm and the gas core. The surface of the liquid ®lm is assumed to be a rough ``pseudo-wall'' (Dobran, 1987; Nigmatulin, 1991). The closure relations are expressed as: If ReF < 1600; tFp
2uF d
and tGF tFp :
1 ; uFG 8=7uF : If ReF r1600; tFp Cw rF u2F ; Cw 0:083Reÿ0:25 F 2 p rF 1=3 1 2 ÿ0:25 tGF CGF rG uG ÿ uFG ; CGF 0:0791ReG 1 24 1 ÿ aG 2 rG 8 where d is the mean averaged thickness (m), rF the liquid ®lm density (kg/m3), uF the averaged liquid ®lm velocity (m/s), u FG the velocity at the interface between the liquid ®lm and the gas core (m/s), aG the averaged gas fraction and ReF the ®lm Reynolds number de®ned as: ReF
4rF uF d mF
9
The heat ¯ux Q_ FG exerted from the liquid ®lm to the gas core is correlated from: Q_ FG h~F TF ÿ TG ;
10
where TF is the averaged ®lm temperature (K), and the heat ¯ux coecient is given by analogy with Colburn's relation: p h~F D aG 0:4 NuF 0:023Re0:8 11 G PrG : lG Figure 3 compares, for the same experimental conditions in Fig. 2, the measured pressure pro®le to PAFM's predictions. The pressure pro®le which is calculated with the measured gas mass ¯owrate (Fig. 3, full line) is obviously beyond the data of pressure. To get closer to the measured pressure pro®le, a very great value of gas mass ¯owrate should be introduced, as shown the pressure pro®le predicted with a higher gas mass ¯owrate equal to 95 kg/h (Fig. 3, intermittent line), whereas the measured value is 65.8 kg/h.
328
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 3. Comparison with measured pressure (symbol) and predictions of PAFM for various gas mass ¯ow-rate values.
The PAFM is also unable, as the two-®eld model, to predict well the experimental results, but it appears clear that the liquid ®lm is a source of high mechanical non-equilibrium, as underlined by Lemonnier and Camelo (1993). The in¯uence of gas mass ¯owrate on the pressure pro®le, illustrated in Fig. 3, gives us the idea that the improvement of the predicted maximum gas mass ¯owrate in Fig. 1 could be possible if we introduce the liquid ®lm ®eld into the ¯ow modeling. In other words, a good description of the experiments presented here requires a complete modeling of the annular dispersed structure of the ¯ow. 2.3. Conclusion By considering the two extreme con®gurations of the annular dispersed ¯ow pattern, the discrepancies between predictions and experimental results are obviously revealed relating to pressure pro®les inside the throat, maximum gas mass ¯owrates, exit droplets velocities and diameters (Figs. 1±3). For the ®rst time, in conformity with the authors Selmer Olsen and Lemonnier, the two-®eld model has showed that, in all the experiments at an upstream pressure of 2 bar, the main nonequilibrium between the liquid and gas phases is of mechanical nature: the temperature of the liquid and gas phases does not dier signi®cantly. With regard to the data, the discrepancies could be the result of there being no account for the liquid ®lm ¯owing along the nozzle walls, as has been suggested by Selmer Olsen and Lemonnier.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
329
The analysis of data with the PAFM allows us to illustrate the important role of the liquid ®lm on the mechanical non-equilibrium between the phases, on the prediction of the maximum gas mass ¯owrate. Then, again in agreement with Selmer Olsen and Lemonnier, it could be possible to bring improvements to the predictions with a more complete description of the annular dispersed structure. Although it has been clear from the beginning that an annular dispersed ¯ow model must be established to describe these experiments in a converging nozzle, this step leads us to the understanding of how an annular or dispersed con®guration has a signi®cant eect on the maximum gas ¯owrate, due to dierent types of momentum transfer between the gas phase and the liquid phase as ®lm or droplets. From here, we have started on the modeling of the annular dispersed ¯ow. Our purpose is to develop a three-®eld model, which not only accounts for frictions and heat ¯ux already introduced in the two-®eld model and the PAFM, but also for the mass transfer between the liquid ®lm and the gas core. Selmer Olsen (1991) has shown the necessity to introduce the liquid deposition and entrainment mechanisms in order to take into account the nonstabilization of the two-phase ¯ow in the nozzle. The main diculty in the complete modeling of annular dispersed two-phase ¯ow, is the increase of the number of closure relations. Consequently, we have intended to reach it step by step by assuming for a start, in the next section, that all the eects due to entrainment and deposition processes are negligible.
3. ``Frozen'' Three-Field Model Our primary approach to the annular dispersed two-phase ¯ow model is to assume negligible the liquid mass transfer, in the mass balance equations as well as in the momentum and energy balance equations. The whole correlations required for the closure model have been de®ned in the previous sections, and the assumptions of the two ¯ow models have also been retained. A distribution of the total liquid mass ¯owrate between the ®lm and the droplets is set upstream of the nozzle. Introducing the measured value of the gas mass ¯owrate, pressure pro®les are calculated with varying ®lm mass ¯owrate or entrained liquid (Figs. 4 and 5). The entrained liquid fraction E is de®ned as the ratio between the liquid mass ¯owrate dragged along the gas core from the ®lm surface, and the total liquid mass ¯owrate. Under the same experimental conditions than in Figs. 2 and 3, Fig. 4 shows that: . If we assume a rather annular gas±liquid con®guration, with a high ®lm mass ¯owrate (equal to 180 kg/h with respect to the total liquid mass ¯owrate of 187.6 kg/h), or a low entrained liquid fraction E of about 0.2, the predicted pressure pro®le is obviously beyond measured points of pressure (Fig. 4, intermittent line): these results are similar to those of PAFM (see Fig. 3).
330
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 4. Comparison with measured pressure (symbol) and predictions of ``Frozen'' three-®eld model for various entrained liquid fraction values.
Fig. 5. Comparison of measured pressure and ®lm thickness (symbols) with predictions of ``Frozen'' three-®eld model for various entrained liquid fraction values. The symbol (q) represents the mean ®lm thickness, the crosses (+) indicate one standard deviation up or down, and the arrow tips (r and t) indicate extreme values.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
331
. If we assume a rather dispersed gas±liquid con®guration, with an important amount of entrained liquid, the predicted pressure drop is far too considerable inside the throat (Fig. 4, mixed line): the results are now similar to those of the two-®eld model (see Fig. 2). . However, with a certain value of entrained liquid (equal to 0.5), the pressure pro®le is well predicted (Fig. 4, full line). Fig. 4 proves the existence of distribution ®lm-droplets of the total liquid mass ¯owrate, with which the pressure pro®le gets close to the measured pro®le. In other words, the entrained liquid fraction at the nozzle inlet has a signi®cant eect on the calculation of the pressure pro®le. It is at the vicinity of the throat inlet that the in¯uence of the entrained liquid fraction value is signi®cant on the prediction of the pressure drop. As the upstream gas±liquid con®guration is annular, a droplet moisture exchange between the liquid ®lm and dispersed droplets can generate the appropriate distribution ®lm-droplets at the neighborhood of the throat inlet. In Fig. 5, the strong in¯uence of the entrained liquid fraction E, is also observed for two other experimental conditions, as well as for the pressure pro®les as on evolution of ®lm thickness. It is noticed that an insucient pressure drop inside the throat occurs for the case of the higher total liquid mass ¯owrate. By assuming the gas mass ¯owrate to be equal to the measured one, we have analyzed the eects of the entrained liquid fraction at the inlet, on the pressure and ®lm thickness pro®les. The great sensitivity with respect to the value of the entrained liquid fraction, is located at the vicinity of the throat inlet. This suggests that an important liquid entrainment occurs along the converging part of the nozzle, since the liquid injection is annular. The growth of the data of ®lm thickness shown in Fig. 5 suggests a liquid deposition along the nozzle throat. The analysis of the data with the ``Frozen'' three-®eld model shows that the liquid mass transfer between the ®lm and the gas core plays a non-negligible role on the evolution of the annular two-phase ¯ow through the converging nozzle. Furthermore, combined with the experimental results, it allows a rough localization of the entrainment and deposition phenomena in the nozzle. From the previous results, we can now go to the next step, which is to investigate the correlations of entrainment and deposition rates in order to complete the three-®eld model presented in the Appendix. This model is a 10-equation, annular dispersed, two-phase ¯ow model, which also accounts for mechanical non-equilibrium as well as thermal non-equilibrium, and liquid mass exchange between the ®lm and the gas core by droplet deposition and entrainment (see the Appendix).
4. With liquid mass exchange correlations from literature In the three-®eld model, the rates of entrainment and deposition, mÇ E and mÇ D (kg/s.m2), respectively, are present in each of the balance equations established for liquid ®lm and dispersed droplets. In addition to closure relations speci®ed for the ``frozen'' three-®eld model, it remains to de®ne the relations of deposition and entrainment rates.
332
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Table 1 Three major categories of existing correlations of mass transfer by entrainment and deposition, expressed from independent ¯ow variables .
Hutchinson and Whalley (1973) and Whalley s and Hewitt (1978): r m_ D kC m2L tGF aL ; u ; C rL ; k 87u ; DsrL rG aL aG m_ E kC1
where equilibrium droplets concentration C 1 depends on tid/s . Hewitt and Govan (1990): r r D 0:18 if C=rG < 0:3 _ mD kC; k G 0:083 C=rG ÿ0:65 if C=rG > 0:3; s 0:316 _FÿM _ 2 DrL _F >M _ ; _ G M m_ E 5:75 10ÿ5 M if M F F sr2G r _ m rF M D with F ReF exp 5:8504 0:4249 G : AmF mF rG . Nigmatulin (1991): _L 4M d 2 rF tGF 0:3 1 ÿ 7:5aL ; pDDh 18mG rG Dh uG 8 c > rG =rL 0:67 < m_ E k c m_ D ; k c 1 ÿ 0:01aÿ0:17 L 0:5 c d m_ E m_ E m_ E with _F r 4M 0:85 > m_ d m sg Reÿ1 0:55 r L : : F WeGF ÿ WeGF E 2 G p DDh m_ D
From a review of existing correlations, three major categories of relations of mass transfer could be distinguished and are listed in Table 1. In each of these categories, the entrainment mechanism is due to the shearing-o of roll wave crest by streaming gas ¯ow. Nigmatulin (1991) has assumed another mechanism, in addition to the preceding one, due to shocks produced by depositing droplets of the ¯ow core striking the ®lm, and referred to as a shock splash break-away, which is correlated as directly proportional to the droplet deposition rate. Figure 6 illustrates the pressure and ®lm mass ¯owrate pro®les calculated from the three®eld model, with each one of the correlations presented in Table 1, in dierent experimental conditions. It had not be possible with the last two models suggested by Hewitt and Govan (1990) and Nigmatulin (1991), to describe the entrainment and the deposition along the converging nozzle: there's almost no variation of ®lm mass ¯owrates (Fig. 6). Furthermore, it had been necessary, as in the ``Frozen'' three-®eld model, to ®x from the channel inlet a ®lm-droplet distribution of the total liquid mass ¯owrate, which is not compatible with the upstream experimental conditions. Nevertheless, the ®rst model of deposition±entrainment according to Whalley and Hewitt (1978) and Hutchinson and Whalley (1973), is the only one which has succeeded to express an important mass exchange between ®lm and gas core in the converging part of the nozzle from a rather annular upstream liquid con®guration, for all the considered experiments (Fig. 6). However, the predicted pressure drops more rapidly before the exit throat and the ®lm ¯owrate goes on decreasing inside the throat: then the entrainment is occurring, whereas Selmer Olsen had noticed, experimentally, a liquid re-deposition on the ®lm surface.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
333
Fig. 6. Comparison of data with predictions of pressure and ®lm mass ¯ow-rate of the three-®eld model from mass transfer correlations according to dierent authors (Hutchinson and Whalley, 1973; Whalley and Hewitt, 1978; Hewitt and Govan, 1990; Nigmatulin, 1991), in experimental conditions performed in a converging nozzle.
It is clear that investigations are necessary to improve the predictions in a converging nozzle. The only correlations which can succeeded in describing the experiments of Lemonnier and Selmer Olsen, are those proposed by Hutchinson and Whalley (1973) and Whalley and Hewitt (1978) who, moreover, had agreed on the necessity to bring improvements to their relation of entrainment rate. More recently, Nigmatulin et al. (1996) have modi®ed their correlations of droplet deposition and entrainment rates using data from steam±water ¯ows. It is conceivable to analyze the data in a converging nozzle with these relations, but from here, we have decided to focus on the simple model of entrainment rate proposed by Hutchinson and Whalley (1973). This is the subject of the next section, where other experimental data are used, which are more speci®c to the liquid mass exchange.
334
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
5. Investigating a simple model of entrainment rate In view of a better understanding of entrainment and deposition mechanisms, we have applied the various mass transfer correlations listed in Table 1, to the experiments performed by Cousins et al. (1965). These authors had measured the entrained liquid fraction E from the ®lm to the gas core, from annular air±water two-phase ¯ow through a channel of constant cross section. The investigations are based on three distinct experimental conditions performed at an upstream pressure equal to 2.76 bar, and varying ®lm Reynolds numbers and gas mass ¯owrates. The experiments of Cousins et al. (1965) had been described successfully with the correlation of entrained liquid fraction E, elaborated by Ishii and Mishima (1981). This one is expressed as following with an exponential relaxation function in the developing ¯ow region, on the one hand, and with a constant E 1 which depends on the entrance conditions in the fully developed region on the other hand: ÿ1:87:10ÿ5 z2 E1 if z<440 12 E 1 ÿ e if zr440, E1 where the equilibrium value of entrained fraction E 1 is given by: E1 tan h 7:25 10ÿ7 We1:25 Re0:25 LF and the dimensionless distance z by: z Weÿ0:25 Re0:5 z LF ; Dh
13
14
with Re LF the total liquid Reynolds number, Dh the pipe hydraulic diameter, z the axial distance from pipe inlet, and We the entrainment Weber number de®ned by: rG j2G Dh rF ÿ rG 1=3 We ; 15 s rG jG is the gas super®cial velocity. The calculations of entrained fraction are illustrated in Fig. 7 (thin full line), showing that a good agreement with experimental data is achieved. Afterwards, these curves will be used as the basis of comparison with local relations of entrainment and deposition rates. Kataoka and Ishii (1983) had also distinguished, in the experiments of Cousins et al. (1965) another region at the near-vicinity to the channel inlet, namely the geometry-dependent inlet region, where the droplets entrainment rate is almost non-existent. As for Lemonnier and Selmer Olsen's experiments, the various mass transfer correlations of Table 1 have been introduced in the three-®eld model. As for the results obtained in the previous section, the two last deposition±entrainment models proposed by Hewitt and Govan (1990) and Nigmatulin (1991) have not given satisfactory predictions: the dierences with data are too large. It is, again the correlations of Whalley and Hewitt (1978) and Hutchinson and Whalley (1973) which allowed the predictions
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
335
Fig. 7. Comparison of data of Cousins et al. (1965) with: (a) entrained fraction predicted from relation developed by Ishii and Mishima (1981); (b) predictions of three-®eld model obtained from various mass transfer correlations (see Table 1).
of E to get close to the data, the predictions-data dierences being more signi®cant for the case of lower ®lm Reynolds number. The conclusions are similar to those for Lemonnier and Selmer Olsen's experiments, strengthening thus our decision to investigate from the deposition and entrainment rate relations of Whalley and Hewitt and Hutchinson and Whalley. In accordance with the opinion of these authors, we have to bring modi®cations to the entrainment rate relation, with a view to improving the predictions. Hutchinson and Whalley (1973) have assumed that the deposition rate is directly proportional to the droplets mass concentration C (kg/m3) in the gas core: m_ D kC;
16
where k is the mass transfer coecient. The entrainment rate is given by a relation identical to that of the deposition rate, but assuming that a hydrodynamic equilibrium is achieved: m_ E kC1 ;
17
where C 1 is the equilibrium droplet concentration. It remains to determine the expression of the equilibrium droplet concentration. The model of Hutchinson and Whalley (1973) characterizes the entrainment mechanism, based on the shearing-o of roll wave crests, as the result of two opposing forces: the interfacial friction exerted at the interface ®lm-gas, which tends to drag droplets from the ®lm
336
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
surface and the surface tension stress, which keeps the liquid to the ®lm surface. The ratio between the two forces is de®ned in terms of the following dimensionless group: We GF = t GFd/s. We have expressed, analytically, the equilibrium droplet concentration by: C1 124We2GF :
18
As improvements to bring to the entrainment rate relation, Whalley and Hewitt (1978) had suggested to take into account the eect due to the formation of disturbance waves on the interface ®lm-gas, which is directly dependent on the wave amplitude. According to Ishii and Mishima (1989), the wave amplitude is proportional to the ®lm Reynolds number. This dimensionless number is precisely a varying parameter in the experiments of Cousins et al. (1965). Consequently, we propose an adjustment from the experiences of Cousins et al., assuming that the multiplier factor in (18) is no longer a constant, but a function of the ®lm number Reynolds. An iterative procedure is utilized to search for the numerical value of the multiplier factor which allows the predictions to approximate better than the experimental results, in dierent experimental conditions. A linear adjustment of the numerical results lead us to the following relation which is valid only for ®lm Reynolds numbers of more than 800 (Ho-Kee-King, 1996): C1 0:06ReF ÿ ReF We2GF with ReF 800:
19
This limitation is similar to a condition of entrainment onset proposed by Dallman (1984), Andreussi (1983) or Schadel et al. (1990). It is one of the three main conditions of entrainment onset available in literature. The two other conditions are the minimum gas velocity de®ned by Ishii and Grolmes (1975), and the critical ®lm Weber number WeGF according to Nigmatulin (1991). The former condition is always veri®ed in the Cousins et al. experiences, and the last one is the only one which is not explicitly taken into account in the proposed entrainment rate relation (17) and (19). Then, we introduced it in the modi®ed relation of Hutchinson and Whalley. The entrainment rate is ®nally formulated as following: for ReF >ReF 800; WeGF >WeGF m_ E kC1 ; C1 0:06ReF ÿ ReF We2GF 20 m_ E 0 for ReF RReF 800; WeGF RWeGF : Figure 8 illustrates the entrained fraction calculated from the preceding relation (Fig. 8, full line). The modi®cation of Hutchinson's relation leads to obvious ameliorations for the two ®rst experimental conditions with lower ®lm Reynolds number. Prediction-data discrepancies still remain when ReLF is equal to 932, but the dierences obtained earlier in Fig. 7 are now reduced by at least a factor of ten. It is also noticed, in Fig. 8, that the entrainment rate is almost zero on a certain distance from the channel inlet, the length of the distance varies with the experimental conditions. This could correspond to the inlet region identi®ed by Kataoka and Ishii (1983). The length suggested by these authors is given for a dimensionless distance z less than 160: this condition had been only veri®ed for relatively high ®lm Reynolds numbers. In the case of higher ®lm
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
337
Fig. 8. Comparison of data of Cousins et al. (1965) with: (a) predicted entrained fraction from relation developed from Ishii and Mishima (1981); (b) predictions of three-®eld model obtained with the modi®cation of the entrainment rate relation of Hutchinson and Whalley (1973).
Reynolds number (ReLF = 3727, see Fig. 8), the entrainment rate is negligible for a dimensionless distance z less than 130: this is comparable with the condition suggested by Kataoka and Ishii (1983). Although it still remains a discrepancy between predictions and data of the entrained fraction in the case of lower ®lm Reynolds number, the entrainment rate correlation modi®ed from Hutchinson's one, allows a rather good description of most of Cousins's experiences. Consequently, we propose the following correlations as the deposition±entrainment model: s 8 r > m2L tGF > < m_ D kC; k 87u ; u DsrL rG > > : m_ E kC1 ; C1 0:06ReF ÿ ReF We2GF for ReF > ReF 800; WeGF > WeGF : 21 To resume the procedure followed progressively until here, our objective is to establish a three-®eld model which accounts for entrainment and deposition mechanisms in order to describe the experiments of Lemonnier and Selmer Olsen (1992) performed in a converging nozzle. As we have not found satisfying mass transfer relations, we are induced to modify the entrainment rate relation of Hutchinson et al. (1974) on the basis of experimental results of Cousins et al. (1965), who have measured the entrained liquid amount from the ®lm surface to the ¯ow core. A local entrainment rate relation, for a ®xed deposition rate correlation, is established satisfactorily for the experiments of Cousins et al. (1965), we now revert to the experiences of Lemonnier and Selmer Olsen in a converging nozzle.
338
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
6. Predictions of the Three-®eld Model in a Converging Nozzle To reach a good description of the experiments of Lemonnier and Selmer Olsen in a converging nozzle from the three-®eld model, in which entrainment and deposition rates had been correlated by (21), we have to modify some closure relations: . on the one hand, the entrainment rate relation given by (20) will be completed with a condition on the ®lm thickness, below which no further entrainment occurs in these experiences; . and, on the other hand, the parietal ®lm-wall friction relation of type Blasius will be modi®ed to account for the eect of streaming gas ¯ow. For three dierent experimental conditions, presented in Fig. 9, with decreasing values of gas quality, the three-®eld model calculations showed, as those obtained with the original relation of Hutchinson et al. (1974), that: . the liquid ®lm is strongly broken away in the converging part of the nozzle, and the ®lm thickness becomes very thin from the throat inlet: this is in conformity to Selmer Olsen's observations. . and, in addition to the fact that the pressure drops too fast before the throat exit, the liquid break-away is going on inside the throat leading to a reduction of the ®lm thickness, whereas Selmer Olsen measures a slight growth of the ®lm thickness due to a liquid re-deposition. So as to describe these experiments, the break-away in the nozzle throat must be limited. Any one of the existing conditions of entrainment onset have lead us to justify such a limitation (Ho-Kee-King, 1996). Then, we suggest to take into account the existence of a singularity in the nozzle's pro®le, due to the presence of a junction between the converging and the throat by the consideration of the following mechanism: . the break-away in the converging part of the nozzle is eective enough to generate a relatively ``smooth'' ®lm ¯ow, with low amplitude waves on the ®lm surface; . and the distance which is necessary for the growing wave to rise again to the shearing-o of the roll wave crest, is not achieved by the throat length. The entrainment rate model in (20) is, therefore, completed, assuming that, in these experimental conditions, the ®lm thickness is too thin to ensure the phenomena of shearing-o of roll wave crest. So, we add this condition: If d < d ; d 100 mm; then m_ E 0:
22
With the droplet entrainment rate correlated by (20) and (22), Fig. 10 shows that the ®lm mass ¯owrate increases along the throat: that is the proof that the break-away is very limited for the three experimental conditions. Moreover, the trend in the evolution of the ®lm thickness is well reproduced.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
339
Fig. 9. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The entrainment rate relation is modi®ed from Hutchinson's one (20). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down and the arrow tips (r and t) indicate extreme values.
Nevertheless, the pressure pro®les show that: . with a high gas quality, the predicted pressure pro®le is obviously beyond the experimental pro®le; . with a low gas quality, the predicted pressure drop is too important in the throat; and . for an intermediate value of the gas quality, the pressure pro®le is well predicted. In order to locate the reasons, we have analyzed, from the total momentum balance, what terms have implied a strong pressure gradient. The total momentum balance is expressed as follows: ÿA
_ dP d Mu P Fp tFp ; dz dz
23
340
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 10. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The entrainment rate relation is given by (20) and (22). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down and the arrow tips (r and t) indicate extreme values.
where P Fp is the pipe perimeter, dP/dz the total pressure gradient, and the rate of change of the total momentum is given by: X d M _ _ k uk d Mu : dz dz kG;F;L
24
In (23), the two main constituents of total pressure gradient are the parietal friction exerted from the ®lm, and the rate of change of total ¯ow momentum. This last term is preponderant in the converging part, with an order of magnitude relatively important, and it has the same order of magnitude with the friction stress in the throat (Fig. 11). A priori, the term which has a signi®cant in¯uence on the pressure drop, only in the nozzle throat, is the parietal friction. Consequently, we have modi®ed the parietal friction relation in order to introduce the in¯uence of gas quality.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
341
Fig. 11. Order of magnitude of the various terms which appear explicitly in the total momentum balance equation (23).
For the considered experimental conditions, the ®lm ¯ow is turbulent and the parietal friction is correlated by (8) (Nigmatulin, 1991). We assume that, as the ®lm thickness is very thin from the throat, the parietal friction is intensi®ed by the streaming gas ¯ow when the gas quality is relatively high. With an adjustment from experimental conditions, the parietal friction coecient is so correlated (Ho-Kee-King, 1996): 1 1 24:5x2 ÿ 6:45x: tFp Cw rF u2F ; Cw 0:0791Reÿ0:25 F 2
25
We have imposed for the adjustment that the parietal friction coecient tends to Blasius's one for a liquid single phase ¯ow when the gas quality tends towards 0. With such a modi®cation, the three-®eld model predicts, at best, the data of pressure and ®lm thickness along the converging nozzle (Fig. 12). A slight insuciency in predicted pressure drop, in the case of lower gas quality, is noticed. The three-®eld model ®nally proposed here is validated successfully with three dierent experimental conditions. Now, we will apply it to the whole of Lemonnier's experiments performed at an upstream pressure of 2 bar, varying total liquid mass ¯owrates. In these experiments, the three-®eld model has been completed for the cases of low liquid mass ¯owrates. It has already been mentioned that in such conditions the two-phase ¯ow is essentially dispersed and the two-®eld model gives good predictions for maximum gas mass ¯owrates and exit droplet velocities. When the liquid ¯owrate is low (less than 50 kg/h), the predicted ®lm thickness is too thin (about 10 mm). Furthermore, the ®lm ¯ow is laminar, so the interfacial friction exerted between
342
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 12. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The parietal friction coecient is modi®ed with a multiplier factor function of gas quality x (25). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down, and the arrow tips (r and t) indicate extreme values.
the ®lm surface and the gas core is very high, increasing in this way the ®lm velocity. In these conditions, with a very thin ®lm thickness and a high ®lm velocity, the singularity in the nozzle shape due to the junction between the convergent and the throat, which has already been underlined previously, leads now to a quasi-total atomization of the ®lm to the gas core from the throat inlet (Ho-Kee-King, 1996). Predictions of the three-®eld model that we ®nally proposed in this paper reproduce rather well all the data for maximum gas mass ¯owrates and exit droplet velocities, at an upstream pressure of 2 bar (Fig. 13). 7. Conclusions We have shown that air±water ¯ows of which the con®guration is annular dispersed, can be predicted well only with a three-®eld model, accounting for entrainment and deposition mechanisms. It has also been observed (Boyer and Lemonnier 1996) that for steam±water ¯ows in a Venturi tube, the pressure drop was better predicted with a three-®eld model.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
343
Fig. 13. Comparison of experimental data (circular symbols) with: (a) predictions of the three-®eld model (checkedsquare symbols); (b) predictions of the two-®eld model (star symbols), for an upstream pressure constant equal to 2 bar, and varying total liquid mass ¯ow-rates.
The objective of this paper is to reach a good description of the experiments of Lemonnier and Camelo (1993) and Selmer Olsen (1991), performed in a converging nozzle with an annular liquid injection. In the case of such liquid injection mode, Selmer Olsen (1991) has observed that the liquid ®lm is violently atomized at the throat inlet. This behavior is well reproduced with the three-®eld model proposed here, and in which we had had to take into account the multidimensional eects due to the change of cross-sectional area in the nozzle: depending on whether the liquid mass ¯owrate is important or not, the liquid ®lm is partially or totally atomized from the throat inlet. Accounting for the eect of the singularity due to the junction convergent-throat, can justify a multidimensional modeling. With the three-®eld model, it has been possible to describe successfully most of the available experimental data: pressure pro®les, maximum gas mass ¯owrate, exit droplet velocities, and ®lm thicknesses. However, some dierences between predictions and the data still remain unexplained: . The predicted droplet size is always underestimated. This disagreement is probably not only due to the drop size correlation. It is conceivable that a phenomenon of coagulation at the nozzle exit occurs on droplets of small size and of velocity close to the gas velocity. . The pressure drop is sometimes slightly insucient inside the throat. This has been more recently highlighted (Lemonnier H., CEA/Grenoble): a detachment of the ®lm ¯owing on the walls should occur at the vicinity of the throat inlet. The instantaneous nature of some processes will probably be to question. Not only as suggested by Selmer Olsen for the drop size correlation, where it will be necessary to consider a process of break-up combined with a certain relaxation time, but also for the entrainment and deposition phenomena: a certain transit time is necessary to allow the droplets broken away from the ®lm surface, thus of velocity close to ®lm velocity, to reach the velocity of droplets ¯owing in the gas core ¯ow, which is close to the gas velocity.
344
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Improvements are planned, with the contribution of data of entrained liquid fraction and liquid ®lm thickness recently collected by Camelo et al. (1995). All these data will serve as a basis for a further re®ning of the three-®eld model, incorporating the mechanisms invoked form our analysis. These mechanisms have also been invoked from some authors (Hutchinson et al., 1974; Azzopardi et al., 1991; Azzopardi, 1992, 1993; Hills, 1997), and their works will constitute a reference for our investigations.
Appendix A A.1. Two-®eld model : totally dispersed two-phase ¯ow model. Fig. A1 d _ MG 0; dz d _ ML 0; dz d _ dP MG uG AG ÿPtGp ÿ P LG tGL ; dz dz d _ dP ML uL AL P LG tGL ; dz dz d _ 1 2 MG HG uG P LG tGL uL ÿ P LG Q_ LG ; dz 2 d _ 1 2 ML HL uL P LG tGL uL ÿ P LG Q_ LG ; dz 2 d dA AG AL ; dz dz
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
345
_ where the subscripts ``G'' and ``L'' refer to the gas and the liquid droplet phase, respectively, M the mass ¯owrate (kg/s), u the area-averaged velocity (m/s), H the area-averaged mass enthalpy (J/kg), P the area-averaged mixture pressure (N/m2), AG the cross-sectional area occupied from the gas phase (m2), AL the cross-sectional area occupied from the dispersed liquid phase (m2), A the pipe cross-sectional area (m2), P the pipe perimeter (m), P LG the perimeter between the gas and the liquid droplets (m), t the friction stress (N/m2), and Q_ the heat ¯ux (J/s.m2). _ G, M _ L , uG, uL, HG, HL and P. The independent variables are the following: M A.2. PAFM: purely annular two-phase ¯ow model. Fig. A2 d _ MG 0; dz d _ MF 0; dz d _ dP MG uG AG ÿP FG tGF ; dz dz d _ dP MF uF AF ÿPtFp P FG tGF ; dz dz d _ 1 2 ÿP FG tGF uFG P FG Q_ FG ; MG HG uG dz 2 d _ 1 2 P FG tGF uFG ÿ P FG Q_ FG ; MF HF uF dz 2 d dA AG AF ; dz dz where the subscripts ``G'' and ``F'' refer to the gas and the liquid ®lm phase, respectively, AF the cross-sectional area occupied from the liquid ®lm phase (m2), and P FG the perimeter between the gas and the liquid ®lm surface (m).
346
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
_ G, M _ F , uG, uF, HG, HF and P. The independent variables are the following: M A.3. Three-®eld model : annular dispersed two-phase ¯ow model. Fig. A3 d _ MG 0; dz d _ MF P FG m_ D ÿ m_ E ; dz d _ ML ÿ P FG m_ D ÿ m_ E ; dz d _ dP MG uG AG ÿ P FG tGF ÿ P LG tGL ; dz dz d _ dP MF uF AF ÿ P pF tFp P FG tGF P FG m_ D uL ÿ P FG m_ E uFG ; dz dz d _ dP ML uL AL P LG tGL ÿ P FG m_ D uL P FG m_ E uFG ; dz dz d _ 1 2 ÿ P FG tGF uFG ÿ P LG tGL uL P LG Q_ LG P FG Q_ FG ; MG HG uG dz 2 d _ 1 2 1 2 1 2 _ _ _ P FG tGF uFG ÿ P FG QFG P FG mD HL uL ÿ P FG mE HF uFG ; MF HF uF dz 2 2 2 d _ 1 2 1 2 1 2 _ _ _ P LG tGL uL ÿ P LG QLG ÿ P LG mD HL uL P FG mE HF uFG ; ML HL uL dz 2 2 2 d dA AG AF AL : dz dz _ G, M _ F, M _ F , uG, uF, uL, HG, HF, HL and P. The independent variables are the following: M
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
347
References Andreussi, P., 1983. Droplet transfer in two-phase annular ¯ow. Int. J. Multiphase ¯ow 9 (6), 697±713. Azzopardi, B.J., Texeira, S.F.C.F., Govan, A.H., Bott, T.R., 1991. An improved model for pressure drop in venturi scrubbers. Trans. IChemE, B 9, 237±245. Azzopardi, B.J., 1992. Gas-liquid ¯ows in cylindrical venturi scrubbers: boundary layer separation in the diuser section. Chem. Engng Sci. 49, 55±64. Azzopardi, B.J., 1993. Liquid distribution in venturi scrubbers: boundary layer separation in the diuser section. Chem. Engng Sci. 48, 2807±2813. Berne, Ph, 1983. Contribution aÁ la mod eÂlisation du taux de production de vapeur par autovaporisation dans les eÂcoulements diphasiques en conduite. TheÁse de Docteur IngeÂnieur, Ecole Centrale des Arts et Manufactures, Paris, France. Bilicky, Z., Dafermes, C., Kestin, J., Majda, G., Zeng, D.L., 1987. Trajectories and singular points in steady-state models of two-phase ¯ows. Int. J. Multiphase Flow 13 (4), 511±533. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Boyer, C., Lemonnier, H., 1996. Design of a ¯ow metering process for two-phase dispersed ¯ows. Int. J. Multiphase Flow 22 (4), 713± 732. Camelo, E., Lemonnier, H., Ochterbeck, J., Selmer, Olsen S., 1995. Characterization of non equilibrium eects on high quality critical ¯ows. Nureth 7 Proc, Saratoga Springs, New York, Sept. 10±15, Rapport NUREG/CP-0142 vol. 1, 111±130. Cousins, L.B., Denton, W.H., Hewitt, G.F., 1965. Liquid mass transfer in annular two-phase ¯ow. Symp. on Two-Phase Flow, Exeter, UK vol. 1, Paper C4. Dallman, J.C., Laurinat, J.E., Hanratty, T.J., 1984. Entrainment for horizontal annular gas±liquid ¯ow. Int. J. Multiphase Flow 10 (6), 677±690. Dobran, F., 1987. Nonequilibrium modeling of two-phase critical ¯ows in tubes. J. Heat Transfer 109, 731±738. Hewitt, G.F., Govan, H., 1990. Phenomenological modelling of non-equilibrium ¯ows with phase change. Int. J. Heat Mass Transfer 33 (2), 229±242. Hills, J.H. 1997. The critical ¯owrates for wave and droplet formation in annular gas±liquid ¯ow. In: Giot, M., Mayinger, F., Celata, GP. (Eds.), Experimental Heat Transfer, Fluid Mechanics and Thermodynamics. Edizione ETS, Italy. vol. 2. pp. 1241±1247. Hutchinson, P., Whalley, P.B., 1973. A possible characterization of entrainment in annular ¯ow. Chem. Engng. Sci. 28, 974±975. Hutchinson, P., Whalley, P.B., Hewitt, G.F., 1974. Transient ¯ow redistribution in annular two-phase ¯ow. Int. J. Multiphase Flow 1, 383±393. Ho-Kee-King, S., 1996. Contribution aÁ la modeÂlisation des eÂcoulements diphasiques en canalisation: in¯uence des processus d'entra|à nement et de deÂposition dans le cas d'une con®guration annulaire disperseÂe. TheÁse de Doctorat de l'Universite Paris XII-Val de Marne, France. Ishii, M., Grolmes, M.A., 1975. Inception criteria for droplet entrainment in two-phase concurrent ®lm ¯ow. AIChE J. 21 (2), 308±318. Ishii, M., Mishima, K., 1981. Correlation for liquid entrainment in annular two-phase ¯ow of low viscous ¯uid. Argonne Laboratory Report, ANL/RAS/LWR 81-2. Ishii, M., Mishima, K., 1989. Droplet entrainment correlation in annular two-phase ¯ow. Int. J. Heat Mass Transfer 32 (10), 1835± 1846. Kataoka, I., Ishii, M. 1983. Entrainment and deposition rates of droplets in annular two-phase ¯ow. In: Mori, Y., Yang, W.J. (Eds.), Proc. ASME-JSME Thermal Engng Joint Conference. ASME, New York, vol. 1. pp. 69±80. Lemonnier, H., Selmer, Olsen S., 1992. Experimental investigation and physical modelling of two-phase two-component ¯ow in a converging-diverging nozzle. Int. J. Multiphase Flow 18 (1), 1±20. Lemonnier, H., Camelo, Cavalcanti E.S., 1993. Droplet size and velocity at the exit of a nozzle with two-component near-critical and critical ¯ow. ANS Proceedings, 1993 National Heat Transfer Conference, American Nuclear Society vol. 7, 93±100. Nigmatulin, R.I., Nigmatulin, B.I., Khodzhaev, Y.A., Kroshilin, V.E., 1996. Entrainment and deposition rates in a dispersed-®lm ¯ow. Int. J. Multiphase Flow 22 (1), 19±30. Nigmatulin, R.I., 1991. Dynamics of multiphase media, vol. 2. Section 7, Hemisphere. Paleev, I.I., Filippovich, B.S., 1966. Phenomena of liquid transfer in two-phase dispersed annular ¯ow. Int. J. Heat Mass Transfer 9, 1089±1093. Richter, H.J., 1983. Separated two-phase ¯ow model: application to critical two-phase ¯ow. Int. J. Multiphase Flow 9 (5), 511±530. Selmer, Olsen S., 1991. Etude theÂorique et expeÂrimentale des eÂcoulements diphasiques en tuyeÁre convergente-divergente. TheÁse de Doctorat de l'Institut National Polytechnique de Grenoble, France. Schadel, S.A., Leman, G.W., Binder, J.L., Hanratty, T.J., 1990. Rates of atomization and deposition in vertical annular ¯ow. Int. J. Multiphase Flow 16 (3), 363±374. Whalley, P.B., Hewitt, G.F., 1978. The correlation of liquid entrainment fraction and entrainment rate in annular two-phase ¯ow, Rept. AERE-R9187. UKAEA, Harwell.
International Journal of Multiphase Flow 25 (1999) 321±347
Eects of entrainment and deposition mechanisms on annular dispersed two-phase ¯ow in a converging nozzle S. Ho Kee King *, G. Piar Laboratoire d'EnergeÂtique et de Thermique Industrielle de l'Est Francilien (LETIEF), Universite Paris XII-Val de Marne, IUT de CreÂteil, Avenue du GeÂneÂral de Gaulle, 94010, CreÂteil CeÂdex, France
Abstract This paper is intended to establish a simple three-®eld model, which gives good agreement with a converging nozzle data previously obtained. The problems encountered in developing a mathematical model for two-phase ¯ow are numerous and generally complex. More particularly, the main diculties concern the closure relations generally correlated empirically and which govern the various transfers at the interfaces and walls. Our methodology consists in taking into account at ®rst the limits of the annular dispersed con®guration before reaching step by step its complete description. Such a procedure underlines the necessity to describe the entrainment and deposition mechanisms in order to get a better annular dispersed two-phase ¯ow understanding. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Two-phase ¯ow; Air-water; Annular dispersed ¯ow; Converging nozzle; One-dimensional model; Closure laws; Entrainment rate; Deposition rate; Entrained liquid fraction; Onset of entrainment
1. Introduction The study of two-phase ¯ows is necessary to evaluate or optimize the performances of many industrials systems currently encountered in chemical, thermal or nuclear engineering. For example, in petroleum industry, where ¯uids are generally in pressurized tanks storage, the security of industrial equipment relies on the correct sizing of the safety valve to prevent for an eventual accidental rupture of a lique®ed tank, or the protection of chemical reactors against thermal runaway. * Corresponding author. Tel.: (33) (1) 45 17 18 41; Fax: (33) (1) 45 17 18 42; E-Mail: [email protected] 0301-9322/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 9 3 2 2 ( 9 8 ) 0 0 0 4 3 - 3
322
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
The purpose of this paper is to investigate how such a ¯ow develops through a converging nozzle. Owing to the present state of knowledge, we are only interested in developing onedimensional (1D) ¯ow models. Droplet moisture exchange is an important factor for mass transfer of annular dispersed two-phase ¯ows. Such a ¯ow is characterized by a central gas core containing droplets ¯owing with a velocity more rapid than the liquid ®lm on the pipe wall. Its complete modeling is based on writing the conservation equations of mass, momentum and energy for each one of the three-®elds : liquid ®lm, gas and dispersed droplets. The problems generally encountered in modeling two-phase ¯ows are in establishing analytically the transfers between the ¯uids under consideration and the walls, on the one hand, and between the ¯uids themselves at the interfaces on the other hand. The laws giving these various transfers allow one to close the conservation equations system and are so called closure laws. Among these laws, the more dicult to specify are the deposition and entrainment rates not yet as well-known as, for example, ®lm-gas, gas-droplets interfacial friction, or wall friction. Our work is essentially based on air±water experiments performed at CEA/Grenoble by Lemonnier and Camelo (1993) and Selmer Olsen (1991) on the FOSSEGRIMEN ¯ow loop, with an annular liquid injection mode. This loop is designed by Selmer Olsen in order to produce annular dispersed critical air±water two-phase ¯ow. The experimental data are collected from a vertical downward converging nozzle with a 10 mm diameter long throat : data of pressure and ®lm thickness pro®les, maximum gas mass ¯owrate, size and velocity of droplets at the nozzle exit. The experimental setup and the techniques of measure are described in detail by Selmer Olsen (1991) and Lemonnier and Camelo (1993). In a ®rst step, two simpli®ed ¯ow models are utilized to analyze the experimental data : totally dispersed ¯ow and purely annular ¯ow models. It appears clearly that neither the liquid ®lm or dispersed droplets can be neglected in Lemonnier and Selmer Olsen's experiments. However, it must be emphasized that when the gas mass quality is very high the ¯ow is basically dispersed; so the two-®eld model predicts quite well the maximum gas mass ¯owrate and exit liquid droplet velocities (Lemonnier and Camelo 1993; Ho-Kee-King 1996). Therefore, the three-®eld model must verify numerical results identical to those of the two-®eld model when the two-phase ¯ow tends to become essentially dispersed. At the beginning, the experiments of Lemonnier and Selmer Olsen have been carried out to characterize the eects of non-equilibrium. These authors have already compared their data with predictions of a dispersed ¯ow model (DFM), which neglects the eect of liquid ®lm presence at walls, and where only the dierence of velocities between the phases is taken into account. With the two-®eld model presented here, which assumes the two-phase ¯ow being totally dispersed, any kind of non-equilibrium (mechanical or thermal) is taken into account. In the considered experiments, predictions of such a model have con®rmed that the main source of non-equilibrium is eectively the mechanical's one and that thermal non-equilibrium between the phases can be neglected. The next step consists in a more complete modeling with the ``frozen'' three-®eld model, the whole eects of droplet entrainment and deposition being neglected. So, for the model closure, the main assumption, which is highly speculative, is that a quasi-equilibrium is established between the annular ®lm, at walls and the dispersed gas core from the nozzle inlet. The
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
323
purpose of such a step in our work is to highlight the eects on pressure pro®le of the entrained liquid fraction at the nozzle inlet, more particularly at the throat inlet. In the presented experiments, comparisons of data with predictions of pressure and ®lm thickness pro®les, showed that droplet moisture exchange between the liquid ®lm and the dispersed central gas core would take place in a large majority in the converging part of the nozzle. After this step, we have intended to give a more detailed description of annular dispersed air±water ¯ows in nozzles with the three-®eld model, incorporating entrainment and deposition mechanisms. For lack of suitable correlations of entrainment and deposition rates in literature, our objective is to search for a deposition±entrainment model which provides a good agreement between the three-®eld model's predictions, with available experimental data obtained in a converging nozzle.
2. Limits con®gurations of annular dispersed two-phase ¯ow modeling The two ¯ow models presented in this section are the totally dispersed ¯ow model (two-®eld model ), where the presence of liquid ®lm is neglected, on the one hand, and the purely annular ¯ow model (PAFM ), assuming negligible the eects of dispersed droplets in the gas core, on the other hand. The thermodynamical properties of the ¯uids are de®ned by assuming that air follows perfect gas laws and water to be an incompressible ¯uid. 2.1. Totally dispersed ¯ow model (two-®eld model) Ho-Kee-King (1996) has presented a 7-equation dispersed ¯ow model, namely the two-®eld model, based on six independent balance equations (one mass balance, one momentum balance and one energy balance for each of the two phases), and one equation which imposes the pipe's pro®le (see the Appendix). This last equation appears explicitly in the system because of the following selected independent variables: _ L ; uG ; uL ; HG ; HL ; P; _ G; M M
1
_ L the dispersed liquid mass ¯owrate (kg/s), uG the _ G is the gas mass ¯owrate (kg/s), M where M averaged gas velocity (m/s), uL the averaged dispersed droplets velocity (m/s), HG the averaged mass enthalpy (J/kg), HL the averaged droplets mass enthalpy (J/kg), and P the mixture pressure (N/m2). This model diers from the 5-equation dispersed ¯ow model (DFM) proposed by Lemonnier and Selmer Olsen (1992), only by the fact that it includes the dierences of temperatures between the phases in addition to the mechanical non-equilibrium. The four closure relations to be provided to the two-®eld model are the wall friction from the gas phase t Gp (N/m2), the interfacial friction between gas and droplets t GL (N/m2), the droplets diameter d (m) and the heat ¯ux at gas-droplet interfaces Q_ LG (J/s.m2). For the ®rst three relations just mentioned, we have utilized those of DFM.
324
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
The wall friction is given by the classical Blasius's relation for a single ¯uid ¯ow in pipe: tGp
1 r uG D fru2 ; f 0:0791Reÿ0:25 ; ReG G ; G 2 mG
2
where r and u are the density (kg/m3) and the velocity (m/s) of two-phase mixture, respectively, D the pipe diameter (m) and mG the gas dynamic viscosity (kg/m.s). The interfacial friction at gas-droplets interfaces proposed by Selmer Olsen is correlated as: 1 r d juG ÿ uL j ; Red G : tGL CD rG uG ÿ uL juG ÿ uL j; CD 6:3Reÿ0:385 d 2 mG
3
The droplet size is calculated from the maximum stable diameter according to Berne (1983): 3=5 3=5 3 ÿ2=5 u 25 1 24 s d 4 2 kR 42 p 2 3rL =rG rL which depends of the friction velocity u de®ned as: p u tGp =r;
5
where k is the Von KaÂrmaÂn constant (10.41), R the pipe radius (m), rL the liquid density, and s the liquid surface tension (N/m). For the last closure relation, the heat ¯ux from the liquid dispersed phase to the gas phase is correlated by (Richter, 1983; Dobran, 1987): Q_ LG h~L TL ÿ TG ;
6
where TL and TG are the averaged temperature of the droplets and the gas, respectively (K ), and the heat ¯ux coecient h~L is given by analogy with a ¯ow surrounding a single sphere (Bird et al., 1960): NuL
h~L d 1=3 2 0:6Re1=2 d PrG ; lG
7
where lG is the gas heat conductivity (W/K.m). Figure 1 presents comparisons of predictions with data of maximum gas mass ¯owrate, droplet velocity, and size at the nozzle exit, for all the experiments performed by Lemonnier and Camelo (1993) when the upstream pressure is set to 2 bar and varying liquid mass ¯owrate. The gas mass quality is ranged between 96 and 25%. We ®rst compared predictions of the two-®eld model proposed here (Fig. 1, starry symbol) with still those of the two-®eld model, but setting a heat ¯ux coecient suciently high so as to reach for an eective heat transfer between the two phases (Fig. 1, squared symbol). These last predictions are obviously identical to DFM's ones. Such a comparison showed equivalent results, so Selmer Olsen's assumption in the DFM is con®rmed : the eects of the thermal non-equilibrium can be neglected for these experiments. It should be noticed that for a liquid mass ¯owrate below 50 kg/h the calculated maximum gas mass ¯owrate is slightly higher when the thermal equilibrium is not imposed.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
325
Fig. 1. Comparison of data (circular symbol) with (a) predictions of two-®eld model (starry symbol) and (b) predictions of two-®eld model but imposing thermal equilibrium between the phases (squared symbol). The upstream pressure is constant and equal to 2 bar.
Next, a comparison of predictions (Fig. 1, starry or squared symbol) with data (Fig. 1, circular symbol) showed, according to the conclusions of Lemonnier and Camelo (1993), that: . with a liquid mass ¯owrate value less than 50 kg/h, the maximum gas mass ¯owrate and exit droplet velocity are well predicted; . for higher liquid mass ¯owrates, the discrepancies increase with the increase of the liquid mass ¯owrate; . the droplet size predicted at the nozzle exit is always smaller than the measured drop size, the dierences become more signi®cant with the increase of the liquid mass ¯owrate. This could be the consequence of a phenomena of drops coalescence occurring from the nozzle outlet and the location where the measurements had been performed (at 15 mm apart from the throat end, Lemonnier and Camelo 1995). If now we look at the case where the gas quality is equal to 25%, or the liquid mass ¯owrate is equal to 187.6 kg/h (see Fig. 1), we have also compared the measured pressure pro®le (Fig. 2, symbol) with the pressure pro®les predicted by the two-®eld model. The ®rst pressure pro®le (Fig. 2, thin full line) is obtained with the predicted critical gas mass ¯owrate, which is calculated iteratively so that the downstream pressure becomes equal to 1 bar: its value of 55.2 kg/h is lower than the experimental value (equal to 65.8 kg/h). Thus, it is remarked that, although the downstream pressure is the atmospheric pressure, the calculated pressure drop inside the throat disagrees with the data. The second pressure pro®le (Fig. 2, intermittent line) is calculated with the measured gas mass ¯owrate (equal to 65.8 kg/h). It can be noticed that, up to a certain section, a good agreement is reached with the data and, downstream, the pressure drops so fast that the predictions present a vertical tangent. If we refer to the detailed analysis of Bilicki et al. (1987) on the topological description of the ensemble of solutions, the point oering a vertical tangent
326
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 2. Comparison with measured pressure (symbol) and predictions of two-®eld model for various gas mass ¯owrate values.
should be a turning point ; so the nature of the two-phase ¯ow is critical and the ¯ow is not feasible. Figure 2 also presents another pressure pro®le obtained with an intermediate value of gas mass ¯owrate, and it illustrates the displacement of the critical section towards the throat exit. The two-®eld model can reproduce successfully the experimental results only when the ¯ow is basically dispersed (with high gas mass quality or low liquid mass ¯owrate). In the experiments at an upstream pressure of 2 bar presented in this section, the discrepancies appear signi®cantly when the liquid ¯owrate is higher than 50 kg/h: the maximum gas mass ¯owrate and the exit droplet diameter and velocity are all underestimated, and the pressure drop inside the nozzle throat is not well reproduced. Moreover, since the presence of ®lm liquid is neglected, the two-®eld model is not able to describe the eect of the annular gas±liquid con®guration at the nozzle inlet. The reasons evoked by Selmer Olsen (1991) and Lemonnier and Camelo (1993) in their analysis, are focused on the fact that the liquid ®lm should play an important role on the acceleration of the gas±liquid ¯ow. Then, this could explain the underestimations of the maximum gas ¯owrate and, in the same way, the in¯uence of the gas mass ¯owrate on the pressure pro®le illustrated in Fig. 2. In order to highlight the eects of the ®lm on the mechanical non-equilibrium between the phases, we have analyzed the data with the other limit ¯ow model's predictions, which assumes all the liquid phase as ®lm on the nozzle walls.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
327
2.2. Purely annular two-phase ¯ow model (PAFM) The liquid injection mode at the nozzle inlet is annular, and it is assumed here that the ¯ow pattern remains the same along the nozzle. The assumptions of the previous ¯ow model are preserved, except that the dispersed liquid phase is now neglected and the ®lm thickness is assumed uniformly distributed all around the pipe wall. The system of the equations of PAFM are presented in the Appendix. Concerning the closure laws, the parietal friction t Fp (N/m2) is exerted from the liquid ®lm and the interfacial friction t GF (N/m2) at the interface between the surface of the liquid ®lm and the gas core. The surface of the liquid ®lm is assumed to be a rough ``pseudo-wall'' (Dobran, 1987; Nigmatulin, 1991). The closure relations are expressed as: If ReF < 1600; tFp
2uF d
and tGF tFp :
1 ; uFG 8=7uF : If ReF r1600; tFp Cw rF u2F ; Cw 0:083Reÿ0:25 F 2 p rF 1=3 1 2 ÿ0:25 tGF CGF rG uG ÿ uFG ; CGF 0:0791ReG 1 24 1 ÿ aG 2 rG 8 where d is the mean averaged thickness (m), rF the liquid ®lm density (kg/m3), uF the averaged liquid ®lm velocity (m/s), u FG the velocity at the interface between the liquid ®lm and the gas core (m/s), aG the averaged gas fraction and ReF the ®lm Reynolds number de®ned as: ReF
4rF uF d mF
9
The heat ¯ux Q_ FG exerted from the liquid ®lm to the gas core is correlated from: Q_ FG h~F TF ÿ TG ;
10
where TF is the averaged ®lm temperature (K), and the heat ¯ux coecient is given by analogy with Colburn's relation: p h~F D aG 0:4 NuF 0:023Re0:8 11 G PrG : lG Figure 3 compares, for the same experimental conditions in Fig. 2, the measured pressure pro®le to PAFM's predictions. The pressure pro®le which is calculated with the measured gas mass ¯owrate (Fig. 3, full line) is obviously beyond the data of pressure. To get closer to the measured pressure pro®le, a very great value of gas mass ¯owrate should be introduced, as shown the pressure pro®le predicted with a higher gas mass ¯owrate equal to 95 kg/h (Fig. 3, intermittent line), whereas the measured value is 65.8 kg/h.
328
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 3. Comparison with measured pressure (symbol) and predictions of PAFM for various gas mass ¯ow-rate values.
The PAFM is also unable, as the two-®eld model, to predict well the experimental results, but it appears clear that the liquid ®lm is a source of high mechanical non-equilibrium, as underlined by Lemonnier and Camelo (1993). The in¯uence of gas mass ¯owrate on the pressure pro®le, illustrated in Fig. 3, gives us the idea that the improvement of the predicted maximum gas mass ¯owrate in Fig. 1 could be possible if we introduce the liquid ®lm ®eld into the ¯ow modeling. In other words, a good description of the experiments presented here requires a complete modeling of the annular dispersed structure of the ¯ow. 2.3. Conclusion By considering the two extreme con®gurations of the annular dispersed ¯ow pattern, the discrepancies between predictions and experimental results are obviously revealed relating to pressure pro®les inside the throat, maximum gas mass ¯owrates, exit droplets velocities and diameters (Figs. 1±3). For the ®rst time, in conformity with the authors Selmer Olsen and Lemonnier, the two-®eld model has showed that, in all the experiments at an upstream pressure of 2 bar, the main nonequilibrium between the liquid and gas phases is of mechanical nature: the temperature of the liquid and gas phases does not dier signi®cantly. With regard to the data, the discrepancies could be the result of there being no account for the liquid ®lm ¯owing along the nozzle walls, as has been suggested by Selmer Olsen and Lemonnier.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
329
The analysis of data with the PAFM allows us to illustrate the important role of the liquid ®lm on the mechanical non-equilibrium between the phases, on the prediction of the maximum gas mass ¯owrate. Then, again in agreement with Selmer Olsen and Lemonnier, it could be possible to bring improvements to the predictions with a more complete description of the annular dispersed structure. Although it has been clear from the beginning that an annular dispersed ¯ow model must be established to describe these experiments in a converging nozzle, this step leads us to the understanding of how an annular or dispersed con®guration has a signi®cant eect on the maximum gas ¯owrate, due to dierent types of momentum transfer between the gas phase and the liquid phase as ®lm or droplets. From here, we have started on the modeling of the annular dispersed ¯ow. Our purpose is to develop a three-®eld model, which not only accounts for frictions and heat ¯ux already introduced in the two-®eld model and the PAFM, but also for the mass transfer between the liquid ®lm and the gas core. Selmer Olsen (1991) has shown the necessity to introduce the liquid deposition and entrainment mechanisms in order to take into account the nonstabilization of the two-phase ¯ow in the nozzle. The main diculty in the complete modeling of annular dispersed two-phase ¯ow, is the increase of the number of closure relations. Consequently, we have intended to reach it step by step by assuming for a start, in the next section, that all the eects due to entrainment and deposition processes are negligible.
3. ``Frozen'' Three-Field Model Our primary approach to the annular dispersed two-phase ¯ow model is to assume negligible the liquid mass transfer, in the mass balance equations as well as in the momentum and energy balance equations. The whole correlations required for the closure model have been de®ned in the previous sections, and the assumptions of the two ¯ow models have also been retained. A distribution of the total liquid mass ¯owrate between the ®lm and the droplets is set upstream of the nozzle. Introducing the measured value of the gas mass ¯owrate, pressure pro®les are calculated with varying ®lm mass ¯owrate or entrained liquid (Figs. 4 and 5). The entrained liquid fraction E is de®ned as the ratio between the liquid mass ¯owrate dragged along the gas core from the ®lm surface, and the total liquid mass ¯owrate. Under the same experimental conditions than in Figs. 2 and 3, Fig. 4 shows that: . If we assume a rather annular gas±liquid con®guration, with a high ®lm mass ¯owrate (equal to 180 kg/h with respect to the total liquid mass ¯owrate of 187.6 kg/h), or a low entrained liquid fraction E of about 0.2, the predicted pressure pro®le is obviously beyond measured points of pressure (Fig. 4, intermittent line): these results are similar to those of PAFM (see Fig. 3).
330
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 4. Comparison with measured pressure (symbol) and predictions of ``Frozen'' three-®eld model for various entrained liquid fraction values.
Fig. 5. Comparison of measured pressure and ®lm thickness (symbols) with predictions of ``Frozen'' three-®eld model for various entrained liquid fraction values. The symbol (q) represents the mean ®lm thickness, the crosses (+) indicate one standard deviation up or down, and the arrow tips (r and t) indicate extreme values.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
331
. If we assume a rather dispersed gas±liquid con®guration, with an important amount of entrained liquid, the predicted pressure drop is far too considerable inside the throat (Fig. 4, mixed line): the results are now similar to those of the two-®eld model (see Fig. 2). . However, with a certain value of entrained liquid (equal to 0.5), the pressure pro®le is well predicted (Fig. 4, full line). Fig. 4 proves the existence of distribution ®lm-droplets of the total liquid mass ¯owrate, with which the pressure pro®le gets close to the measured pro®le. In other words, the entrained liquid fraction at the nozzle inlet has a signi®cant eect on the calculation of the pressure pro®le. It is at the vicinity of the throat inlet that the in¯uence of the entrained liquid fraction value is signi®cant on the prediction of the pressure drop. As the upstream gas±liquid con®guration is annular, a droplet moisture exchange between the liquid ®lm and dispersed droplets can generate the appropriate distribution ®lm-droplets at the neighborhood of the throat inlet. In Fig. 5, the strong in¯uence of the entrained liquid fraction E, is also observed for two other experimental conditions, as well as for the pressure pro®les as on evolution of ®lm thickness. It is noticed that an insucient pressure drop inside the throat occurs for the case of the higher total liquid mass ¯owrate. By assuming the gas mass ¯owrate to be equal to the measured one, we have analyzed the eects of the entrained liquid fraction at the inlet, on the pressure and ®lm thickness pro®les. The great sensitivity with respect to the value of the entrained liquid fraction, is located at the vicinity of the throat inlet. This suggests that an important liquid entrainment occurs along the converging part of the nozzle, since the liquid injection is annular. The growth of the data of ®lm thickness shown in Fig. 5 suggests a liquid deposition along the nozzle throat. The analysis of the data with the ``Frozen'' three-®eld model shows that the liquid mass transfer between the ®lm and the gas core plays a non-negligible role on the evolution of the annular two-phase ¯ow through the converging nozzle. Furthermore, combined with the experimental results, it allows a rough localization of the entrainment and deposition phenomena in the nozzle. From the previous results, we can now go to the next step, which is to investigate the correlations of entrainment and deposition rates in order to complete the three-®eld model presented in the Appendix. This model is a 10-equation, annular dispersed, two-phase ¯ow model, which also accounts for mechanical non-equilibrium as well as thermal non-equilibrium, and liquid mass exchange between the ®lm and the gas core by droplet deposition and entrainment (see the Appendix).
4. With liquid mass exchange correlations from literature In the three-®eld model, the rates of entrainment and deposition, mÇ E and mÇ D (kg/s.m2), respectively, are present in each of the balance equations established for liquid ®lm and dispersed droplets. In addition to closure relations speci®ed for the ``frozen'' three-®eld model, it remains to de®ne the relations of deposition and entrainment rates.
332
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Table 1 Three major categories of existing correlations of mass transfer by entrainment and deposition, expressed from independent ¯ow variables .
Hutchinson and Whalley (1973) and Whalley s and Hewitt (1978): r m_ D kC m2L tGF aL ; u ; C rL ; k 87u ; DsrL rG aL aG m_ E kC1
where equilibrium droplets concentration C 1 depends on tid/s . Hewitt and Govan (1990): r r D 0:18 if C=rG < 0:3 _ mD kC; k G 0:083 C=rG ÿ0:65 if C=rG > 0:3; s 0:316 _FÿM _ 2 DrL _F >M _ ; _ G M m_ E 5:75 10ÿ5 M if M F F sr2G r _ m rF M D with F ReF exp 5:8504 0:4249 G : AmF mF rG . Nigmatulin (1991): _L 4M d 2 rF tGF 0:3 1 ÿ 7:5aL ; pDDh 18mG rG Dh uG 8 c > rG =rL 0:67 < m_ E k c m_ D ; k c 1 ÿ 0:01aÿ0:17 L 0:5 c d m_ E m_ E m_ E with _F r 4M 0:85 > m_ d m sg Reÿ1 0:55 r L : : F WeGF ÿ WeGF E 2 G p DDh m_ D
From a review of existing correlations, three major categories of relations of mass transfer could be distinguished and are listed in Table 1. In each of these categories, the entrainment mechanism is due to the shearing-o of roll wave crest by streaming gas ¯ow. Nigmatulin (1991) has assumed another mechanism, in addition to the preceding one, due to shocks produced by depositing droplets of the ¯ow core striking the ®lm, and referred to as a shock splash break-away, which is correlated as directly proportional to the droplet deposition rate. Figure 6 illustrates the pressure and ®lm mass ¯owrate pro®les calculated from the three®eld model, with each one of the correlations presented in Table 1, in dierent experimental conditions. It had not be possible with the last two models suggested by Hewitt and Govan (1990) and Nigmatulin (1991), to describe the entrainment and the deposition along the converging nozzle: there's almost no variation of ®lm mass ¯owrates (Fig. 6). Furthermore, it had been necessary, as in the ``Frozen'' three-®eld model, to ®x from the channel inlet a ®lm-droplet distribution of the total liquid mass ¯owrate, which is not compatible with the upstream experimental conditions. Nevertheless, the ®rst model of deposition±entrainment according to Whalley and Hewitt (1978) and Hutchinson and Whalley (1973), is the only one which has succeeded to express an important mass exchange between ®lm and gas core in the converging part of the nozzle from a rather annular upstream liquid con®guration, for all the considered experiments (Fig. 6). However, the predicted pressure drops more rapidly before the exit throat and the ®lm ¯owrate goes on decreasing inside the throat: then the entrainment is occurring, whereas Selmer Olsen had noticed, experimentally, a liquid re-deposition on the ®lm surface.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
333
Fig. 6. Comparison of data with predictions of pressure and ®lm mass ¯ow-rate of the three-®eld model from mass transfer correlations according to dierent authors (Hutchinson and Whalley, 1973; Whalley and Hewitt, 1978; Hewitt and Govan, 1990; Nigmatulin, 1991), in experimental conditions performed in a converging nozzle.
It is clear that investigations are necessary to improve the predictions in a converging nozzle. The only correlations which can succeeded in describing the experiments of Lemonnier and Selmer Olsen, are those proposed by Hutchinson and Whalley (1973) and Whalley and Hewitt (1978) who, moreover, had agreed on the necessity to bring improvements to their relation of entrainment rate. More recently, Nigmatulin et al. (1996) have modi®ed their correlations of droplet deposition and entrainment rates using data from steam±water ¯ows. It is conceivable to analyze the data in a converging nozzle with these relations, but from here, we have decided to focus on the simple model of entrainment rate proposed by Hutchinson and Whalley (1973). This is the subject of the next section, where other experimental data are used, which are more speci®c to the liquid mass exchange.
334
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
5. Investigating a simple model of entrainment rate In view of a better understanding of entrainment and deposition mechanisms, we have applied the various mass transfer correlations listed in Table 1, to the experiments performed by Cousins et al. (1965). These authors had measured the entrained liquid fraction E from the ®lm to the gas core, from annular air±water two-phase ¯ow through a channel of constant cross section. The investigations are based on three distinct experimental conditions performed at an upstream pressure equal to 2.76 bar, and varying ®lm Reynolds numbers and gas mass ¯owrates. The experiments of Cousins et al. (1965) had been described successfully with the correlation of entrained liquid fraction E, elaborated by Ishii and Mishima (1981). This one is expressed as following with an exponential relaxation function in the developing ¯ow region, on the one hand, and with a constant E 1 which depends on the entrance conditions in the fully developed region on the other hand: ÿ1:87:10ÿ5 z2 E1 if z<440 12 E 1 ÿ e if zr440, E1 where the equilibrium value of entrained fraction E 1 is given by: E1 tan h 7:25 10ÿ7 We1:25 Re0:25 LF and the dimensionless distance z by: z Weÿ0:25 Re0:5 z LF ; Dh
13
14
with Re LF the total liquid Reynolds number, Dh the pipe hydraulic diameter, z the axial distance from pipe inlet, and We the entrainment Weber number de®ned by: rG j2G Dh rF ÿ rG 1=3 We ; 15 s rG jG is the gas super®cial velocity. The calculations of entrained fraction are illustrated in Fig. 7 (thin full line), showing that a good agreement with experimental data is achieved. Afterwards, these curves will be used as the basis of comparison with local relations of entrainment and deposition rates. Kataoka and Ishii (1983) had also distinguished, in the experiments of Cousins et al. (1965) another region at the near-vicinity to the channel inlet, namely the geometry-dependent inlet region, where the droplets entrainment rate is almost non-existent. As for Lemonnier and Selmer Olsen's experiments, the various mass transfer correlations of Table 1 have been introduced in the three-®eld model. As for the results obtained in the previous section, the two last deposition±entrainment models proposed by Hewitt and Govan (1990) and Nigmatulin (1991) have not given satisfactory predictions: the dierences with data are too large. It is, again the correlations of Whalley and Hewitt (1978) and Hutchinson and Whalley (1973) which allowed the predictions
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
335
Fig. 7. Comparison of data of Cousins et al. (1965) with: (a) entrained fraction predicted from relation developed by Ishii and Mishima (1981); (b) predictions of three-®eld model obtained from various mass transfer correlations (see Table 1).
of E to get close to the data, the predictions-data dierences being more signi®cant for the case of lower ®lm Reynolds number. The conclusions are similar to those for Lemonnier and Selmer Olsen's experiments, strengthening thus our decision to investigate from the deposition and entrainment rate relations of Whalley and Hewitt and Hutchinson and Whalley. In accordance with the opinion of these authors, we have to bring modi®cations to the entrainment rate relation, with a view to improving the predictions. Hutchinson and Whalley (1973) have assumed that the deposition rate is directly proportional to the droplets mass concentration C (kg/m3) in the gas core: m_ D kC;
16
where k is the mass transfer coecient. The entrainment rate is given by a relation identical to that of the deposition rate, but assuming that a hydrodynamic equilibrium is achieved: m_ E kC1 ;
17
where C 1 is the equilibrium droplet concentration. It remains to determine the expression of the equilibrium droplet concentration. The model of Hutchinson and Whalley (1973) characterizes the entrainment mechanism, based on the shearing-o of roll wave crests, as the result of two opposing forces: the interfacial friction exerted at the interface ®lm-gas, which tends to drag droplets from the ®lm
336
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
surface and the surface tension stress, which keeps the liquid to the ®lm surface. The ratio between the two forces is de®ned in terms of the following dimensionless group: We GF = t GFd/s. We have expressed, analytically, the equilibrium droplet concentration by: C1 124We2GF :
18
As improvements to bring to the entrainment rate relation, Whalley and Hewitt (1978) had suggested to take into account the eect due to the formation of disturbance waves on the interface ®lm-gas, which is directly dependent on the wave amplitude. According to Ishii and Mishima (1989), the wave amplitude is proportional to the ®lm Reynolds number. This dimensionless number is precisely a varying parameter in the experiments of Cousins et al. (1965). Consequently, we propose an adjustment from the experiences of Cousins et al., assuming that the multiplier factor in (18) is no longer a constant, but a function of the ®lm number Reynolds. An iterative procedure is utilized to search for the numerical value of the multiplier factor which allows the predictions to approximate better than the experimental results, in dierent experimental conditions. A linear adjustment of the numerical results lead us to the following relation which is valid only for ®lm Reynolds numbers of more than 800 (Ho-Kee-King, 1996): C1 0:06ReF ÿ ReF We2GF with ReF 800:
19
This limitation is similar to a condition of entrainment onset proposed by Dallman (1984), Andreussi (1983) or Schadel et al. (1990). It is one of the three main conditions of entrainment onset available in literature. The two other conditions are the minimum gas velocity de®ned by Ishii and Grolmes (1975), and the critical ®lm Weber number WeGF according to Nigmatulin (1991). The former condition is always veri®ed in the Cousins et al. experiences, and the last one is the only one which is not explicitly taken into account in the proposed entrainment rate relation (17) and (19). Then, we introduced it in the modi®ed relation of Hutchinson and Whalley. The entrainment rate is ®nally formulated as following: for ReF >ReF 800; WeGF >WeGF m_ E kC1 ; C1 0:06ReF ÿ ReF We2GF 20 m_ E 0 for ReF RReF 800; WeGF RWeGF : Figure 8 illustrates the entrained fraction calculated from the preceding relation (Fig. 8, full line). The modi®cation of Hutchinson's relation leads to obvious ameliorations for the two ®rst experimental conditions with lower ®lm Reynolds number. Prediction-data discrepancies still remain when ReLF is equal to 932, but the dierences obtained earlier in Fig. 7 are now reduced by at least a factor of ten. It is also noticed, in Fig. 8, that the entrainment rate is almost zero on a certain distance from the channel inlet, the length of the distance varies with the experimental conditions. This could correspond to the inlet region identi®ed by Kataoka and Ishii (1983). The length suggested by these authors is given for a dimensionless distance z less than 160: this condition had been only veri®ed for relatively high ®lm Reynolds numbers. In the case of higher ®lm
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
337
Fig. 8. Comparison of data of Cousins et al. (1965) with: (a) predicted entrained fraction from relation developed from Ishii and Mishima (1981); (b) predictions of three-®eld model obtained with the modi®cation of the entrainment rate relation of Hutchinson and Whalley (1973).
Reynolds number (ReLF = 3727, see Fig. 8), the entrainment rate is negligible for a dimensionless distance z less than 130: this is comparable with the condition suggested by Kataoka and Ishii (1983). Although it still remains a discrepancy between predictions and data of the entrained fraction in the case of lower ®lm Reynolds number, the entrainment rate correlation modi®ed from Hutchinson's one, allows a rather good description of most of Cousins's experiences. Consequently, we propose the following correlations as the deposition±entrainment model: s 8 r > m2L tGF > < m_ D kC; k 87u ; u DsrL rG > > : m_ E kC1 ; C1 0:06ReF ÿ ReF We2GF for ReF > ReF 800; WeGF > WeGF : 21 To resume the procedure followed progressively until here, our objective is to establish a three-®eld model which accounts for entrainment and deposition mechanisms in order to describe the experiments of Lemonnier and Selmer Olsen (1992) performed in a converging nozzle. As we have not found satisfying mass transfer relations, we are induced to modify the entrainment rate relation of Hutchinson et al. (1974) on the basis of experimental results of Cousins et al. (1965), who have measured the entrained liquid amount from the ®lm surface to the ¯ow core. A local entrainment rate relation, for a ®xed deposition rate correlation, is established satisfactorily for the experiments of Cousins et al. (1965), we now revert to the experiences of Lemonnier and Selmer Olsen in a converging nozzle.
338
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
6. Predictions of the Three-®eld Model in a Converging Nozzle To reach a good description of the experiments of Lemonnier and Selmer Olsen in a converging nozzle from the three-®eld model, in which entrainment and deposition rates had been correlated by (21), we have to modify some closure relations: . on the one hand, the entrainment rate relation given by (20) will be completed with a condition on the ®lm thickness, below which no further entrainment occurs in these experiences; . and, on the other hand, the parietal ®lm-wall friction relation of type Blasius will be modi®ed to account for the eect of streaming gas ¯ow. For three dierent experimental conditions, presented in Fig. 9, with decreasing values of gas quality, the three-®eld model calculations showed, as those obtained with the original relation of Hutchinson et al. (1974), that: . the liquid ®lm is strongly broken away in the converging part of the nozzle, and the ®lm thickness becomes very thin from the throat inlet: this is in conformity to Selmer Olsen's observations. . and, in addition to the fact that the pressure drops too fast before the throat exit, the liquid break-away is going on inside the throat leading to a reduction of the ®lm thickness, whereas Selmer Olsen measures a slight growth of the ®lm thickness due to a liquid re-deposition. So as to describe these experiments, the break-away in the nozzle throat must be limited. Any one of the existing conditions of entrainment onset have lead us to justify such a limitation (Ho-Kee-King, 1996). Then, we suggest to take into account the existence of a singularity in the nozzle's pro®le, due to the presence of a junction between the converging and the throat by the consideration of the following mechanism: . the break-away in the converging part of the nozzle is eective enough to generate a relatively ``smooth'' ®lm ¯ow, with low amplitude waves on the ®lm surface; . and the distance which is necessary for the growing wave to rise again to the shearing-o of the roll wave crest, is not achieved by the throat length. The entrainment rate model in (20) is, therefore, completed, assuming that, in these experimental conditions, the ®lm thickness is too thin to ensure the phenomena of shearing-o of roll wave crest. So, we add this condition: If d < d ; d 100 mm; then m_ E 0:
22
With the droplet entrainment rate correlated by (20) and (22), Fig. 10 shows that the ®lm mass ¯owrate increases along the throat: that is the proof that the break-away is very limited for the three experimental conditions. Moreover, the trend in the evolution of the ®lm thickness is well reproduced.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
339
Fig. 9. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The entrainment rate relation is modi®ed from Hutchinson's one (20). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down and the arrow tips (r and t) indicate extreme values.
Nevertheless, the pressure pro®les show that: . with a high gas quality, the predicted pressure pro®le is obviously beyond the experimental pro®le; . with a low gas quality, the predicted pressure drop is too important in the throat; and . for an intermediate value of the gas quality, the pressure pro®le is well predicted. In order to locate the reasons, we have analyzed, from the total momentum balance, what terms have implied a strong pressure gradient. The total momentum balance is expressed as follows: ÿA
_ dP d Mu P Fp tFp ; dz dz
23
340
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 10. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The entrainment rate relation is given by (20) and (22). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down and the arrow tips (r and t) indicate extreme values.
where P Fp is the pipe perimeter, dP/dz the total pressure gradient, and the rate of change of the total momentum is given by: X d M _ _ k uk d Mu : dz dz kG;F;L
24
In (23), the two main constituents of total pressure gradient are the parietal friction exerted from the ®lm, and the rate of change of total ¯ow momentum. This last term is preponderant in the converging part, with an order of magnitude relatively important, and it has the same order of magnitude with the friction stress in the throat (Fig. 11). A priori, the term which has a signi®cant in¯uence on the pressure drop, only in the nozzle throat, is the parietal friction. Consequently, we have modi®ed the parietal friction relation in order to introduce the in¯uence of gas quality.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
341
Fig. 11. Order of magnitude of the various terms which appear explicitly in the total momentum balance equation (23).
For the considered experimental conditions, the ®lm ¯ow is turbulent and the parietal friction is correlated by (8) (Nigmatulin, 1991). We assume that, as the ®lm thickness is very thin from the throat, the parietal friction is intensi®ed by the streaming gas ¯ow when the gas quality is relatively high. With an adjustment from experimental conditions, the parietal friction coecient is so correlated (Ho-Kee-King, 1996): 1 1 24:5x2 ÿ 6:45x: tFp Cw rF u2F ; Cw 0:0791Reÿ0:25 F 2
25
We have imposed for the adjustment that the parietal friction coecient tends to Blasius's one for a liquid single phase ¯ow when the gas quality tends towards 0. With such a modi®cation, the three-®eld model predicts, at best, the data of pressure and ®lm thickness along the converging nozzle (Fig. 12). A slight insuciency in predicted pressure drop, in the case of lower gas quality, is noticed. The three-®eld model ®nally proposed here is validated successfully with three dierent experimental conditions. Now, we will apply it to the whole of Lemonnier's experiments performed at an upstream pressure of 2 bar, varying total liquid mass ¯owrates. In these experiments, the three-®eld model has been completed for the cases of low liquid mass ¯owrates. It has already been mentioned that in such conditions the two-phase ¯ow is essentially dispersed and the two-®eld model gives good predictions for maximum gas mass ¯owrates and exit droplet velocities. When the liquid ¯owrate is low (less than 50 kg/h), the predicted ®lm thickness is too thin (about 10 mm). Furthermore, the ®lm ¯ow is laminar, so the interfacial friction exerted between
342
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Fig. 12. Comparison of measured pressure and ®lm thickness pro®les (symbols) with predictions of the three-®eld model. The parietal friction coecient is modi®ed with a multiplier factor function of gas quality x (25). The symbol (q) represents the mean ®lm thicknesses, the crosses (+) indicate one standard deviation up or down, and the arrow tips (r and t) indicate extreme values.
the ®lm surface and the gas core is very high, increasing in this way the ®lm velocity. In these conditions, with a very thin ®lm thickness and a high ®lm velocity, the singularity in the nozzle shape due to the junction between the convergent and the throat, which has already been underlined previously, leads now to a quasi-total atomization of the ®lm to the gas core from the throat inlet (Ho-Kee-King, 1996). Predictions of the three-®eld model that we ®nally proposed in this paper reproduce rather well all the data for maximum gas mass ¯owrates and exit droplet velocities, at an upstream pressure of 2 bar (Fig. 13). 7. Conclusions We have shown that air±water ¯ows of which the con®guration is annular dispersed, can be predicted well only with a three-®eld model, accounting for entrainment and deposition mechanisms. It has also been observed (Boyer and Lemonnier 1996) that for steam±water ¯ows in a Venturi tube, the pressure drop was better predicted with a three-®eld model.
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
343
Fig. 13. Comparison of experimental data (circular symbols) with: (a) predictions of the three-®eld model (checkedsquare symbols); (b) predictions of the two-®eld model (star symbols), for an upstream pressure constant equal to 2 bar, and varying total liquid mass ¯ow-rates.
The objective of this paper is to reach a good description of the experiments of Lemonnier and Camelo (1993) and Selmer Olsen (1991), performed in a converging nozzle with an annular liquid injection. In the case of such liquid injection mode, Selmer Olsen (1991) has observed that the liquid ®lm is violently atomized at the throat inlet. This behavior is well reproduced with the three-®eld model proposed here, and in which we had had to take into account the multidimensional eects due to the change of cross-sectional area in the nozzle: depending on whether the liquid mass ¯owrate is important or not, the liquid ®lm is partially or totally atomized from the throat inlet. Accounting for the eect of the singularity due to the junction convergent-throat, can justify a multidimensional modeling. With the three-®eld model, it has been possible to describe successfully most of the available experimental data: pressure pro®les, maximum gas mass ¯owrate, exit droplet velocities, and ®lm thicknesses. However, some dierences between predictions and the data still remain unexplained: . The predicted droplet size is always underestimated. This disagreement is probably not only due to the drop size correlation. It is conceivable that a phenomenon of coagulation at the nozzle exit occurs on droplets of small size and of velocity close to the gas velocity. . The pressure drop is sometimes slightly insucient inside the throat. This has been more recently highlighted (Lemonnier H., CEA/Grenoble): a detachment of the ®lm ¯owing on the walls should occur at the vicinity of the throat inlet. The instantaneous nature of some processes will probably be to question. Not only as suggested by Selmer Olsen for the drop size correlation, where it will be necessary to consider a process of break-up combined with a certain relaxation time, but also for the entrainment and deposition phenomena: a certain transit time is necessary to allow the droplets broken away from the ®lm surface, thus of velocity close to ®lm velocity, to reach the velocity of droplets ¯owing in the gas core ¯ow, which is close to the gas velocity.
344
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
Improvements are planned, with the contribution of data of entrained liquid fraction and liquid ®lm thickness recently collected by Camelo et al. (1995). All these data will serve as a basis for a further re®ning of the three-®eld model, incorporating the mechanisms invoked form our analysis. These mechanisms have also been invoked from some authors (Hutchinson et al., 1974; Azzopardi et al., 1991; Azzopardi, 1992, 1993; Hills, 1997), and their works will constitute a reference for our investigations.
Appendix A A.1. Two-®eld model : totally dispersed two-phase ¯ow model. Fig. A1 d _ MG 0; dz d _ ML 0; dz d _ dP MG uG AG ÿPtGp ÿ P LG tGL ; dz dz d _ dP ML uL AL P LG tGL ; dz dz d _ 1 2 MG HG uG P LG tGL uL ÿ P LG Q_ LG ; dz 2 d _ 1 2 ML HL uL P LG tGL uL ÿ P LG Q_ LG ; dz 2 d dA AG AL ; dz dz
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
345
_ where the subscripts ``G'' and ``L'' refer to the gas and the liquid droplet phase, respectively, M the mass ¯owrate (kg/s), u the area-averaged velocity (m/s), H the area-averaged mass enthalpy (J/kg), P the area-averaged mixture pressure (N/m2), AG the cross-sectional area occupied from the gas phase (m2), AL the cross-sectional area occupied from the dispersed liquid phase (m2), A the pipe cross-sectional area (m2), P the pipe perimeter (m), P LG the perimeter between the gas and the liquid droplets (m), t the friction stress (N/m2), and Q_ the heat ¯ux (J/s.m2). _ G, M _ L , uG, uL, HG, HL and P. The independent variables are the following: M A.2. PAFM: purely annular two-phase ¯ow model. Fig. A2 d _ MG 0; dz d _ MF 0; dz d _ dP MG uG AG ÿP FG tGF ; dz dz d _ dP MF uF AF ÿPtFp P FG tGF ; dz dz d _ 1 2 ÿP FG tGF uFG P FG Q_ FG ; MG HG uG dz 2 d _ 1 2 P FG tGF uFG ÿ P FG Q_ FG ; MF HF uF dz 2 d dA AG AF ; dz dz where the subscripts ``G'' and ``F'' refer to the gas and the liquid ®lm phase, respectively, AF the cross-sectional area occupied from the liquid ®lm phase (m2), and P FG the perimeter between the gas and the liquid ®lm surface (m).
346
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
_ G, M _ F , uG, uF, HG, HF and P. The independent variables are the following: M A.3. Three-®eld model : annular dispersed two-phase ¯ow model. Fig. A3 d _ MG 0; dz d _ MF P FG m_ D ÿ m_ E ; dz d _ ML ÿ P FG m_ D ÿ m_ E ; dz d _ dP MG uG AG ÿ P FG tGF ÿ P LG tGL ; dz dz d _ dP MF uF AF ÿ P pF tFp P FG tGF P FG m_ D uL ÿ P FG m_ E uFG ; dz dz d _ dP ML uL AL P LG tGL ÿ P FG m_ D uL P FG m_ E uFG ; dz dz d _ 1 2 ÿ P FG tGF uFG ÿ P LG tGL uL P LG Q_ LG P FG Q_ FG ; MG HG uG dz 2 d _ 1 2 1 2 1 2 _ _ _ P FG tGF uFG ÿ P FG QFG P FG mD HL uL ÿ P FG mE HF uFG ; MF HF uF dz 2 2 2 d _ 1 2 1 2 1 2 _ _ _ P LG tGL uL ÿ P LG QLG ÿ P LG mD HL uL P FG mE HF uFG ; ML HL uL dz 2 2 2 d dA AG AF AL : dz dz _ G, M _ F, M _ F , uG, uF, uL, HG, HF, HL and P. The independent variables are the following: M
S. Ho Kee King, G. Piar / International Journal of Multiphase Flow 25 (1999) 321±347
347
References Andreussi, P., 1983. Droplet transfer in two-phase annular ¯ow. Int. J. Multiphase ¯ow 9 (6), 697±713. Azzopardi, B.J., Texeira, S.F.C.F., Govan, A.H., Bott, T.R., 1991. An improved model for pressure drop in venturi scrubbers. Trans. IChemE, B 9, 237±245. Azzopardi, B.J., 1992. Gas-liquid ¯ows in cylindrical venturi scrubbers: boundary layer separation in the diuser section. Chem. Engng Sci. 49, 55±64. Azzopardi, B.J., 1993. Liquid distribution in venturi scrubbers: boundary layer separation in the diuser section. Chem. Engng Sci. 48, 2807±2813. Berne, Ph, 1983. Contribution aÁ la mod eÂlisation du taux de production de vapeur par autovaporisation dans les eÂcoulements diphasiques en conduite. TheÁse de Docteur IngeÂnieur, Ecole Centrale des Arts et Manufactures, Paris, France. Bilicky, Z., Dafermes, C., Kestin, J., Majda, G., Zeng, D.L., 1987. Trajectories and singular points in steady-state models of two-phase ¯ows. Int. J. Multiphase Flow 13 (4), 511±533. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. Wiley, New York. Boyer, C., Lemonnier, H., 1996. Design of a ¯ow metering process for two-phase dispersed ¯ows. Int. J. Multiphase Flow 22 (4), 713± 732. Camelo, E., Lemonnier, H., Ochterbeck, J., Selmer, Olsen S., 1995. Characterization of non equilibrium eects on high quality critical ¯ows. Nureth 7 Proc, Saratoga Springs, New York, Sept. 10±15, Rapport NUREG/CP-0142 vol. 1, 111±130. Cousins, L.B., Denton, W.H., Hewitt, G.F., 1965. Liquid mass transfer in annular two-phase ¯ow. Symp. on Two-Phase Flow, Exeter, UK vol. 1, Paper C4. Dallman, J.C., Laurinat, J.E., Hanratty, T.J., 1984. Entrainment for horizontal annular gas±liquid ¯ow. Int. J. Multiphase Flow 10 (6), 677±690. Dobran, F., 1987. Nonequilibrium modeling of two-phase critical ¯ows in tubes. J. Heat Transfer 109, 731±738. Hewitt, G.F., Govan, H., 1990. Phenomenological modelling of non-equilibrium ¯ows with phase change. Int. J. Heat Mass Transfer 33 (2), 229±242. Hills, J.H. 1997. The critical ¯owrates for wave and droplet formation in annular gas±liquid ¯ow. In: Giot, M., Mayinger, F., Celata, GP. (Eds.), Experimental Heat Transfer, Fluid Mechanics and Thermodynamics. Edizione ETS, Italy. vol. 2. pp. 1241±1247. Hutchinson, P., Whalley, P.B., 1973. A possible characterization of entrainment in annular ¯ow. Chem. Engng. Sci. 28, 974±975. Hutchinson, P., Whalley, P.B., Hewitt, G.F., 1974. Transient ¯ow redistribution in annular two-phase ¯ow. Int. J. Multiphase Flow 1, 383±393. Ho-Kee-King, S., 1996. Contribution aÁ la modeÂlisation des eÂcoulements diphasiques en canalisation: in¯uence des processus d'entra|à nement et de deÂposition dans le cas d'une con®guration annulaire disperseÂe. TheÁse de Doctorat de l'Universite Paris XII-Val de Marne, France. Ishii, M., Grolmes, M.A., 1975. Inception criteria for droplet entrainment in two-phase concurrent ®lm ¯ow. AIChE J. 21 (2), 308±318. Ishii, M., Mishima, K., 1981. Correlation for liquid entrainment in annular two-phase ¯ow of low viscous ¯uid. Argonne Laboratory Report, ANL/RAS/LWR 81-2. Ishii, M., Mishima, K., 1989. Droplet entrainment correlation in annular two-phase ¯ow. Int. J. Heat Mass Transfer 32 (10), 1835± 1846. Kataoka, I., Ishii, M. 1983. Entrainment and deposition rates of droplets in annular two-phase ¯ow. In: Mori, Y., Yang, W.J. (Eds.), Proc. ASME-JSME Thermal Engng Joint Conference. ASME, New York, vol. 1. pp. 69±80. Lemonnier, H., Selmer, Olsen S., 1992. Experimental investigation and physical modelling of two-phase two-component ¯ow in a converging-diverging nozzle. Int. J. Multiphase Flow 18 (1), 1±20. Lemonnier, H., Camelo, Cavalcanti E.S., 1993. Droplet size and velocity at the exit of a nozzle with two-component near-critical and critical ¯ow. ANS Proceedings, 1993 National Heat Transfer Conference, American Nuclear Society vol. 7, 93±100. Nigmatulin, R.I., Nigmatulin, B.I., Khodzhaev, Y.A., Kroshilin, V.E., 1996. Entrainment and deposition rates in a dispersed-®lm ¯ow. Int. J. Multiphase Flow 22 (1), 19±30. Nigmatulin, R.I., 1991. Dynamics of multiphase media, vol. 2. Section 7, Hemisphere. Paleev, I.I., Filippovich, B.S., 1966. Phenomena of liquid transfer in two-phase dispersed annular ¯ow. Int. J. Heat Mass Transfer 9, 1089±1093. Richter, H.J., 1983. Separated two-phase ¯ow model: application to critical two-phase ¯ow. Int. J. Multiphase Flow 9 (5), 511±530. Selmer, Olsen S., 1991. Etude theÂorique et expeÂrimentale des eÂcoulements diphasiques en tuyeÁre convergente-divergente. TheÁse de Doctorat de l'Institut National Polytechnique de Grenoble, France. Schadel, S.A., Leman, G.W., Binder, J.L., Hanratty, T.J., 1990. Rates of atomization and deposition in vertical annular ¯ow. Int. J. Multiphase Flow 16 (3), 363±374. Whalley, P.B., Hewitt, G.F., 1978. The correlation of liquid entrainment fraction and entrainment rate in annular two-phase ¯ow, Rept. AERE-R9187. UKAEA, Harwell.